Uniform Stabilization and Asymptotic Behavior with a Lower Bound of the Maximal Existence Time of a Coupled System’s Semi-Linear Pseudo-Parabolic Equations

Abstract

This article discusses the initial boundary value problem for a class of coupled systems of semi-linear pseudo-parabolic equations on a bounded smooth domain. Global solutions with exponential decay and asymptotic behavior are obtained when the maximal existence time has a lower bound for both low and overcritical energy cases. A sharp condition linking these phenomena is derived, and it is demonstrated that global existence also applies to the case of the potential well family.

1. Introduction

This study considers the global existence with exponential decay and the asymptotic behavior of solutions with a lower bound on the highest existence duration for cases of both low and overcritical energy regarding a coupled system of a nonlinear pseudo-parabolic equation. Initially, the boundary value problems were conditioned as follows:

$$\left\{\begin{array}{cc}{\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{u}-\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}}={\mathrm{F}}_{\mathrm{u}}(\mathrm{u},\mathrm{v}),& \mathrm{i}\mathrm{n}\hspace{1em}\mathsf{\Omega }\times (0,\mathrm{\infty }),\\ {\mathrm{v}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{v}-\mathsf{\Delta }{\mathrm{v}}_{\mathrm{t}}={\mathrm{F}}_{\mathrm{v}}(\mathrm{u},\mathrm{v}),& \mathrm{i}\mathrm{n}\hspace{1em}\mathsf{\Omega }\times (0,\mathrm{\infty }),\\ \mathrm{u}=0,\mathrm{v}=0,& \mathrm{o}\mathrm{n}\hspace{1em}\mathsf{\Gamma }\times (0,\mathrm{\infty }),\\ \mathrm{u}(\mathrm{x},0)={\mathrm{u}}_0,\mathrm{v}(\mathrm{x},0)={\mathrm{v}}_0,& \mathrm{x}\in \mathsf{\Omega },\end{array}\right.$$

(1)

where ${\mathrm{u}}_0,{\mathrm{v}}_0\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right),\mathsf{\Omega }$ represents a finite domain in ${\mathbb{R}}^{\mathrm{n}}\left(\mathrm{n}\ge 3\right)$ with a smooth boundary $\partial \mathsf{\Omega }=\mathsf{\Gamma }$, and $\mathrm{F}:{\mathbb{R}}^2\to \mathbb{R}$ is a ${\mathrm{C}}^1$-mapping represented as follows:

$$\mathrm{F}(\mathrm{u},\mathrm{v})=\mathsf{\alpha }|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}+2\mathsf{\beta }|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}},$$

(2)

where $1<\mathrm{p}<{\displaystyle \frac{\mathrm{n}+2}{\mathrm{n}-2}},\mathsf{\alpha }>1$ and $\mathsf{\beta }>0$. In addition,

$$\begin{gathered} {\mathrm{f}}_1\left(\mathrm{u},\mathrm{v}\right):={\mathrm{F}}_{\mathrm{u}}\left(\mathrm{u},\mathrm{v}\right)=\left(\mathrm{p}+1\right)\left[\mathsf{\alpha }|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}-1}\left(\mathrm{u}+\mathrm{v}\right)+\mathsf{\beta }|\mathrm{u}{|}^{{\displaystyle \frac{\mathrm{p}-3}{2}}}|\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{u}\right], \\ {\mathrm{f}}_2\left(\mathrm{u},\mathrm{v}\right):={\mathrm{F}}_{\mathrm{v}}\left(\mathrm{u},\mathrm{v}\right)=\left(\mathrm{p}+1\right)\left[\mathsf{\alpha }|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}-1}\left(\mathrm{u}+\mathrm{v}\right)+\mathsf{\beta }|\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}-3}{2}}}|\mathrm{u}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{u}\right], \end{gathered}$$

(3)

$$\mathrm{u}{\mathrm{f}}_1\left(\mathrm{u},\mathrm{v}\right)+\mathrm{v}{\mathrm{f}}_2\left(\mathrm{u},\mathrm{v}\right)=\left(\mathrm{p}+1\right)\mathrm{F}\left(\mathrm{u},\mathrm{v}\right)\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \left(\mathrm{u},\mathrm{v}\right)\in {\mathbb{R}}^2.$$

Equations of this nature effectively model a variety of important physical phenomena, such as the unidirectional propagation of non-linear dispersive long waves [1,2], the seepage of homogeneous fluids through fissured rock [3], and the discrepancy between conductive and thermodynamic temperatures [4].

These equations can be classified as Sobolev-type differential equations, which are characterized by the inclusion of mixed time and space derivatives among the highest-order terms. When a single time derivative is the highest-order term, this classification is termed a pseudo-parabolic equation, which is a concept originating from Showalter’s foundational work in the 1970s [5]. Since that time, numerous compelling findings concerning general models of pseudo-parabolic equations have emerged. A recent overview of studies pertaining to pseudo-parabolic equations was compiled by Cao and Yin [6].

The initial boundary value problem of semi-linear heat equations can be given by the following equation:

$${\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{u}=\mathrm{f}\left(\mathrm{u}\right),$$

where $\mathrm{f}\left(\mathrm{u}\right)$ takes the form of a polynomial such as $|\mathrm{u}{|}^{\mathrm{p}-1}$ or $|\mathrm{u}{|}^{\mathrm{p}}$ with $\mathrm{p}>1$. On this basis, Payne and Sattinger [7] introduced the potential well method (PWM) and demonstrated the global existence, as well as the finite time blow-up, of weak solutions. Liu and Zhao [8] later refined this method, achieving a threshold result that delineates the conditions for both the global existence and non-existence of solutions, including the vacuum isolation of these solutions.

The available outcomes regarding the initially conditioned boundary value problem are recalled using the following single semi-linear pseudo-parabolic equation:

$${\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{u}-\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}}=\mathrm{f}\left(\mathrm{u}\right).$$

(4)

In the specific case where $\mathrm{f}\left(\mathrm{u}\right)={\mathrm{u}}^{\mathrm{p}}$ with $\mathrm{p}>1$, the maximum principle was established by Benedetto and Pierre [9]. Additionally, Cao et al. [10] conducted a detailed study of the associated Cauchy problems, identifying both the critical global existence exponent and the critical Fujita exponent, using both integral representations and the contraction-mapping principle. They demonstrated that solutions can experience a blow-up in finite time when $\mathrm{p}\in (1,1+{\displaystyle \frac{2}{\mathrm{n}}})$.

The PWM was utilized by Xu et al. [11] to analyze Equation (4). They demonstrated the conditions for global existence with exponential decay and the non-existence of solutions when the initial energy $\mathrm{J}\left({\mathrm{u}}_0\right)\le \mathrm{d}$ for the boundary value problems described by Equation (4) and $\mathrm{f}\left(\mathrm{u}\right)={\mathrm{u}}^{\mathrm{p}}$. Specifically, for $1<\mathrm{p}<\mathrm{\infty }$, the results are applicable if $\mathrm{n}=\mathrm{1,2}$, or if $1<\mathrm{p}\le {\displaystyle \frac{\mathrm{n}+2}{\mathrm{n}-2}}$ when $\mathrm{n}\ge 3$. Furthermore, a finite time blow-up occurs when the initial energy $\mathrm{J}\left({\mathrm{u}}_0\right)>\mathrm{d}$, as indicated by the comparison principle.

In a notable contribution [12], Luo employed a differential inequality technique to derive a lower bound on the blow-up time for Equation (4), with $\mathrm{f}\left(\mathrm{u}\right)={\mathrm{u}}^{\mathrm{p}}$, based on specific conditions of the power $\mathrm{p}$ and the initial value. The blasting criterion and the upper bound of the blasting time under certain conditions were established, and the non-blasting behavior and exponential attenuation under different criteria were discussed.

Furthermore, numerous results have emerged addressing the asymptotic behavior of pseudo-parabolic equations with source terms [13], variable index pseudo-parabolic equations [14], and degenerate singular pseudo-parabolic equations [15], all leveraging the PWM.

In their exploration of semi-linear reaction–diffusion systems, Escobedo and Herrero [16] investigated the Cauchy problem and the related initial boundary value problems given by the following equations:

$$\left\{\begin{array}{c}{\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{u}={\mathrm{v}}^{\mathrm{p}},\\ {\mathrm{v}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{v}={\mathrm{u}}^{\mathrm{q}}.\end{array}\right.$$

They established conditions under which solutions either exist globally over time or experience blow-up within a finite time frame, depending upon specific parameters for $\mathrm{p}$ and $\mathrm{q}$.

In related work, Yang et al. [17] studied a nonlinear pseudo-parabolic system defined by the Cauchy problem:

$$\left\{\begin{array}{c}{\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{u}-\mathrm{\alpha }\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}}={\mathrm{v}}^{\mathrm{p}},\\ {\mathrm{v}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{v}-\mathrm{\alpha }\mathsf{\Delta }{\mathrm{v}}_{\mathrm{t}}={\mathrm{u}}^{\mathrm{q}},\end{array}\right.$$

where $\mathrm{p},\mathrm{q}>1$ and $\mathrm{p}\mathrm{q}>1$, incorporating third-order viscous terms. They first identified the critical Fujita exponent and then determined a second critical exponent to characterize the decay rate of the initial data in the coexistence region of both global and non-global solutions. Additionally, they derived time–decay profiles for the global solutions.

More recently, Xu et al. [18] tackled the following initial boundary value problem:

$$\left\{\begin{array}{c}{\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{u}=\left(\right|\mathrm{u}{|}^{2\mathrm{p}}+\left|\mathrm{v}{|}^{\mathrm{p}+1}\right|\mathrm{u}{|}^{\mathrm{p}-1})\mathrm{u},\\ {\mathrm{v}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{v}=\left(\right|\mathrm{v}{|}^{2\mathrm{p}}+\left|\mathrm{u}{|}^{\mathrm{p}+1}\right|\mathrm{v}{|}^{\mathrm{p}-1})\mathrm{v},\end{array}\right.$$

where they identified the global existence, long-term behavior, and finite time blow-up of solutions at both low and critical initial energy levels. For solutions with high energy levels, they established a comparison principle for the parabolic system, which produced insights into both global existence and finite time blow-up.

Ngoc et al. [19] investigated a system of pseudo-parabolic equations involving a memory term under Robin–Dirichlet conditions. They established the local existence and uniqueness of a weak solution using the Faedo–Galerkin method and demonstrated the global existence and decay of weak solutions with suitable initial data. Employing the concavity method, they derived blow-up results for solutions with both non-negative and negative initial energy, establishing both upper and lower bounds for the lifespans of the equations in the context of blow-up times.

While numerous publications have investigated the properties of solutions to single parabolic and pseudo-parabolic equations, research on systems that encompass both of these equation types remains scarce. Notably, nonlinear pseudo-parabolic equations in coupled system forms have garnered little attention. This article aims to further the examination of nonlinear scaled pseudo-parabolic equations, as outlined in Equation (1).

Building on insights from previous studies and employing the potential well method, we explore the invariance of specific sets under the flow of these equations, as well as the vacuum-isolating behavior of the solutions. Our findings validate the global existence of solutions exhibiting exponential decay and clarify their asymptotic behavior. Moreover, we establish a lower bound for the maximal existence time in scenarios characterized by both low initial energy and critical energy levels. A pivotal aspect of our discoveries is the identification of a specific condition that plays a crucial role in these phenomena. Additionally, we demonstrate that our results concerning global existence can also be extended to the potential well family.

The incorporation of pseudo-parabolic terms is fundamental in determining the lifespan, blasting set, and blasting rate of these equations. A thorough understanding of their distinct influences is essential for analyzing the asymptotic behavior of solutions, underscoring the theoretical importance of this research area.

Our results significantly advance the understanding of coupled pseudo-parabolic systems and their long-term dynamics, which is an area that continues to be both active and challenging in the fields of mathematical analysis and applied mathematics. The novel methodologies and findings presented in this study have the potential to encourage further explorations in related fields.

First, we present a description of the weak solution:

Definition 1.

(Weak solution (WS)). Let $(\mathrm{u},\mathrm{v})=(\mathrm{u}\left(\mathrm{x},\mathrm{t}\right),\mathrm{v}\left(\mathrm{x},\mathrm{t}\right)$, denoting Equation (1)’s WS for $\mathrm{\Omega }\times [0,\mathrm{T})$ when $0<\mathrm{T}\le +\infty$ denotes the highest existence duration, if $\mathrm{u},\mathrm{v}\in {\mathrm{L}}^{\infty }(0,\mathrm{T};{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right))$ with ${\mathrm{u}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}\in {\mathrm{L}}^2(0,\mathrm{T};{\mathrm{L}}^2\left(\mathrm{\Omega }\right))$ and Equation (1) is satisfied distributionally, i.e.,

$$\begin{array}{cc}{({\mathrm{u}}_{\mathrm{t}},{\mathrm{w}}_{ })}_2+{(\nabla {\mathrm{u}}_{\mathrm{t}},\nabla \mathrm{w})}_2+{(\nabla \mathrm{u},\nabla \mathrm{w})}_2& ={\left({\mathrm{F}}_{\mathrm{u}}\right(\mathrm{u},\mathrm{v}),\mathrm{w})}_2,\\ {({\mathrm{v}}_{\mathrm{t}},\mathrm{w})}_2+{(\nabla {\mathrm{v}}_{\mathrm{t}},\nabla \mathrm{w})}_2+{(\nabla \mathrm{v},\nabla \mathrm{w})}_2& ={\left({\mathrm{F}}_{\mathrm{v}}\right(\mathrm{u},\mathrm{v}),\mathrm{w})}_2,\end{array}$$

(5)

for any $\mathrm{w}\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right),\mathrm{t}\in (0,\mathrm{T})$ with $\mathrm{u}(\mathrm{x},0)={\mathrm{u}}_0,\mathrm{v}(\mathrm{x},0)={\mathrm{v}}_0$ for $\mathrm{x}\in \mathrm{\Omega }$ and ${(·,·)}_2$ shows the inner product (IP) ${(·,·)}_{{\mathrm{L}}^2\left(\mathrm{\Omega }\right)}.$

Suppose that $\mathrm{H}={\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$ shows the Hilbert space equipped by the general IP. The norm $\Vert (\mathrm{u},\mathrm{v}){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}$ is represented equivalently by the following norm:

$$\Vert (\mathrm{u},\mathrm{v})\Vert ={\left({\int }_{\mathsf{\Omega }}\left[|\nabla \mathrm{u}\left(\mathrm{x}\right){|}^2+|\nabla \mathrm{v}\left(\mathrm{x}\right){|}^2\right]\mathrm{d}\mathrm{x}\right)}^{{\displaystyle \frac{1}{2}}},$$

since it is implied by the Poincaré inequality. The equation ${\mathrm{L}}^2\left(\mathsf{\Omega }\right)\times {\mathrm{L}}^2\left(\mathsf{\Omega }\right)$, whereby

$$\Vert (\mathrm{u},\mathrm{v}){\Vert }_2={\left({\int }_{\mathsf{\Omega }}\left[|\mathrm{u}\left(\mathrm{x}\right){|}^2+|\mathrm{v}\left(\mathrm{x}\right){|}^2\right]\mathrm{d}\mathrm{x}\right)}^{{\displaystyle \frac{1}{2}}}$$

is also used to denote the norm.

Problem (1) possesses a variational structure. Namely, for $(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$, Equation (1)’s energy functional is as follows:

$$\mathrm{E}\left(\mathrm{u},\mathrm{v}\right)={\displaystyle \frac{1}{2}}{\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}-\mathsf{\alpha }{\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}-2\mathsf{\beta }{\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}.$$

(6)

Remark 1.

The energy conservation is satisfied by the definition of the weak solution:

$${\int }_0^{\mathrm{t}}\Vert ({\partial }_{\mathsf{\tau }}\mathrm{u},{\partial }_{\mathsf{\tau }}\mathrm{v}){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }+\mathrm{E}\left(\mathrm{u},\mathrm{v}\right)=\mathrm{E}\left({\mathrm{u}}_0,{\mathrm{v}}_0\right),\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r} \mathrm{t} \ge 0.$$

(7)

For $(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$, the Nehari functional is defined by

$$\mathrm{K}(\mathrm{u},\mathrm{v})={\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}-\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}-2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}.$$

(8)

From the embedding ${\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\hookrightarrow {\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)$ and the Poincaré inequality, we can deduce the following equation:

$${\mathrm{C}}_{\mathrm{*}}=\sup \left\{{\displaystyle \frac{\Vert \mathsf{\phi }{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}}{\Vert \nabla \mathsf{\phi }{\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}}};\mathsf{\phi }\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right),\mathsf{\phi }\ne 0\right\}.$$

(9)

The primary outcomes are presented, where

$$\mathrm{d}:=\mathrm{i}\mathrm{n}\mathrm{f}\left\{{\mathrm{s}\mathrm{u}\mathrm{p}}_{\mathsf{\lambda }\ge 0}\mathrm{E}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})\mid (\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\backslash \left\{\left(0,0\right)\right\}\right\}.$$

(10)

Equation (1)’s solution is related to exponentially decayed global existence when $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)\ge 0$, since the low and overcritical initial cases for energy are a concern.

Theorem 1.

Presume that $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{d}$ and $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)\ge 0$. A global weak solution is admitted by Equation (1) $\mathrm{u},\mathrm{v}\in {\mathrm{L}}^{\infty }(0,\infty ;{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right))$ with ${\mathrm{u}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}\in {\mathrm{L}}^2(0,\infty ;{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right))$, which satisfies the following:

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1(\mathrm{\Omega })}\le {\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0)\Vert }_{{\mathrm{H}}_0^1(\mathrm{\Omega })}{\mathrm{e}}^{{\displaystyle \frac{1}{2}}-(1-\mathrm{\sigma }){\mathrm{d}}_{\mathrm{\Omega }}^{-2}\mathrm{t}}, \mathrm{f}\mathrm{o}\mathrm{r} 0\le \mathrm{t}<\infty ,$$

where $\mathrm{\sigma }=2^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathrm{\alpha }+\mathrm{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}{\left({\displaystyle \frac{\mathrm{p}+1}{\mathrm{p}-1}}\mathrm{d}\right)}^{{\displaystyle \frac{\mathrm{p}-1}{2}}}$ and ${\mathrm{d}}_{\mathrm{\Omega }}$ is the diameter of the bounded domain $\mathrm{\Omega }$.

The following result determines Equation (1)’s lower bound solution for the highest existence duration and shows the asymptotic behavior when the low and overcritical energy case is initially shown as $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)<0$.

Theorem 2.

Presuming that $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}$ and $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)<0$, the maximal existence time $\mathrm{T}$ is bounded below by

$$\mathrm{T}\ge {\displaystyle \frac{1}{\mathrm{\mu }}}{‖{\mathrm{u}}_0,{\mathrm{v}}_0‖}_{{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)}^{-\left(\mathrm{p}-1\right)},$$

where $\mathrm{\mu }=2^{\mathrm{p}-1}\left({\mathrm{p}}^2-1\right)\left(2^{\mathrm{p}}\mathrm{\alpha }+\mathrm{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}$, and Equation (1)’s weak solution satisfies

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)}\le {({\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0)\Vert }_{{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)}^{-(\mathrm{p}-1)}-\mathrm{\mu }\mathrm{t})}^{-{\displaystyle \frac{1}{\mathrm{p}-1}}}<+\infty ,$$

for $0<\mathrm{t}<{\displaystyle \frac{1}{\mathrm{\mu }}}{\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0)\Vert }_{{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)}^{-(\mathrm{p}-1)}$.

Moreover, for all $0<\mathrm{t}<\mathrm{T}$, Equation (1)’s weak solution satisfies

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)}\ge {\left(\mathrm{ϵ}\mathrm{t}+\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0){\Vert }_{{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)}^2\right)}^{{\displaystyle \frac{1}{2}}},$$

where $0<\mathrm{ϵ}<\mathit{min}\{{\displaystyle \frac{1}{2}}(\mathrm{d}-\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)),-\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)\}$.

If $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{d}$ and $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)<0$, the result follows from the choice of $\mathrm{t}={\mathrm{t}}_0$ for a sufficiently small ${\mathrm{t}}_0>0$ instead of $\mathrm{t}=0$, the conditions’ initial time for both $\mathrm{t}$ and $\mathrm{ϵ}$.

The subsequent sections of this article are organized as follows: Section 2 presents the preliminary results that are essential for proving Theorems 1 and 2, which we achieve in Section 3; this is the primary focus of this article. The conclusion is presented in Section 4.

2. Preliminaries

A class of potential wells relative to Equation (1) is introduced and the relevant features are proven in Section 2.1. Some sets’ invariant features, such as Equation (1)’s flow and the behavior of vacuum isolation with the utilization of PWM, are discussed in Section 2.2.

2.1. The Introduction of a Class of Potential Wells

Regarding Equation (1)’s family of potential wells, several features are presented.

Initially, $\mathrm{E}(\mathrm{u},\mathrm{v})$ and $\mathrm{K}(\mathrm{u},\mathrm{v})$, referred to as functionals, are delineated using Equations (6) and (8). Subsequently, functional features are presented.

Lemma 1.

Let $(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\backslash \left\{(0, 0)\right\}$. Then, we have the following:

(i) 

${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathrm{\lambda }\to 0}\mathrm{E}(\mathrm{\lambda }\mathrm{u},\mathrm{\lambda }\mathrm{v})=0$, and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathrm{\lambda }\to +\infty }\mathrm{E}(\mathrm{\lambda }\mathrm{u},\mathrm{\lambda }\mathrm{v})=-\infty$.

(ii) 

${\mathrm{\lambda }}^{\mathrm{*}}={\mathrm{\lambda }}^{\mathrm{*}}(\mathrm{u},\mathrm{v})>0$ uniquely exists such that  ${\left.{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{\lambda }}}\mathrm{E}(\mathrm{\lambda }\mathrm{u},\mathrm{\lambda }\mathrm{v})\right|}_{\mathrm{\lambda }={\mathrm{\lambda }}^{\mathrm{*}}}=0$.

(iii) 

$\mathrm{E}(\mathrm{\lambda }\mathrm{u},\mathrm{\lambda }\mathrm{v})$ increases strictly when $0\le \mathrm{\lambda }<{\mathrm{\lambda }}^{\mathrm{*}}$, decreases strictly when $\mathrm{\lambda }>{\mathrm{\lambda }}^{\mathrm{*}}$, and reaches the highest value at $\mathrm{\lambda }={\mathrm{\lambda }}^{\mathrm{*}}$.

(iv) 

$\mathrm{K}(\mathrm{\lambda }\mathrm{u},\mathrm{\lambda }\mathrm{v})>0$ for $0<\mathrm{\lambda }<{\mathrm{\lambda }}^{\mathrm{*}}$, $\mathrm{K}(\mathrm{\lambda }\mathrm{u},\mathrm{\lambda }\mathrm{v})<0$ for $\mathrm{\lambda }>{\mathrm{\lambda }}^{\mathrm{*}}$ and $\mathrm{K}({\mathrm{\lambda }}^{\mathrm{*}}\mathrm{u},{\mathrm{\lambda }}^{\mathrm{*}}\mathrm{v})=0$, that is,  ${\mathrm{\lambda }}^{\mathrm{*}}=1$.

Proof. 

(i) Let $(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\backslash \left\{(0,0)\right\}$. From the definition of $\mathrm{E}(\mathrm{u},\mathrm{v})$, we obtain

$$\mathrm{E}\left(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v}\right)={\displaystyle \frac{{\mathsf{\lambda }}^2}{2}}{\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}-\mathsf{\alpha }{\mathsf{\lambda }}^{\mathrm{p}+1}{\int }_{\mathsf{\Omega }}\left|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}-2\mathsf{\beta }{\mathsf{\lambda }}^{\mathrm{p}+1}{\int }_{\mathsf{\Omega }}\right|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x},$$

which gives ${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathsf{\lambda }\to 0}\mathrm{E}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})=0$ and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathsf{\lambda }\to +\mathrm{\infty }}\mathrm{E}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})=-\mathrm{\infty }$.

(ii) An easy calculation shows that

$$\begin{gathered} {\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\lambda }}}\mathrm{E}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})=\mathsf{\lambda }[{\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}-\mathsf{\alpha }\left(\mathrm{p}+1\right){\mathsf{\lambda }}^{\mathrm{p}-1}{\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x} \\ -2\mathsf{\beta }\left(\mathrm{p}+1\right){\mathsf{\lambda }}^{\mathrm{p}-1}{\int }_{\mathsf{\Omega }}\left|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\right] \end{gathered}$$

(11)

(iii) Via a straightforward computation, (11) leads to $(\mathrm{u},\mathrm{v})\ne (0,0)$ where

$${\mathsf{\lambda }}^{\mathrm{*}}={[{\displaystyle \frac{{\int }_{\mathsf{\Omega }}\left(\right|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2)\mathrm{d}\mathrm{x}}{(\mathrm{p}+1){\int }_{\mathsf{\Omega }}\left(\mathsf{\alpha }\right|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}+2\mathsf{\beta }\left|\mathrm{u}\mathrm{v}{|}^{\frac{\mathrm{p}+1}{2}}\right)\mathrm{d}\mathrm{x}}}]}^{{\displaystyle \frac{1}{\mathrm{p}-1}}}>0$$

exists uniquely such that ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\lambda }}}\mathrm{E}({\mathsf{\lambda }}^{\mathrm{*}}\mathrm{u},{\mathsf{\lambda }}^{\mathrm{*}}\mathrm{v})=0$. Moreover,

$$\begin{gathered} {\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\lambda }}}\mathrm{E}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})>0 \mathrm{for} 0<\mathsf{\lambda }<{\mathsf{\lambda }}^{\mathrm{*}}, \\ {\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\lambda }}}\mathrm{E}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})<0 \mathrm{for} {\mathsf{\lambda }}^{\mathrm{*}}<\mathsf{\lambda }<\mathrm{\infty }. \end{gathered}$$

Thus, (iii) holds.

(iv)

$$\begin{gathered} \mathrm{K}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})={\mathsf{\lambda }}^2{\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}-\mathsf{\alpha }\left(\mathrm{p}+1\right){\mathsf{\lambda }}^{\mathrm{p}+1}{\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x} \\ -2\mathsf{\beta }\left(\mathrm{p}+1\right){\mathsf{\lambda }}^{\mathrm{p}+1}{\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x} \\ =\mathsf{\lambda }{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathsf{\lambda }}}\mathrm{E}(\mathsf{\lambda }\mathrm{u},\mathsf{\lambda }\mathrm{v})\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em} \end{gathered}$$

is attained by the conclusion. □

The Nehari manifold is defined as follows:

$$\mathcal{N}=\left\{\right(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1(\mathsf{\Omega })\times {\mathrm{H}}_0^1(\mathsf{\Omega }\left)\right|\mathrm{K}\left(\mathrm{u},\mathrm{v}\right)=0,(\mathrm{u},\mathrm{v})\ne (0,0)\}.$$

Thus, the description of $\mathrm{d}\left(10\right)$ and Lemma 1 suggest that

$$\mathrm{d}={\mathrm{i}\mathrm{n}\mathrm{f}}_{\left(\mathrm{u},\mathrm{v}\right)\in \mathcal{N}} \mathrm{E}(\mathrm{u},\mathrm{v})>0.$$

(12)

The potential well (PW) related to Equation (1) leads to the following set:

$$\mathrm{W}=\left\{\right(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1(\mathsf{\Omega })\times {\mathrm{H}}_0^1(\mathsf{\Omega }\left)\right|\mathrm{E}(\mathrm{u},\mathrm{v})<\mathrm{d},\mathrm{K}(\mathrm{u},\mathrm{v})>0\}\cup \{\left(\mathrm{0,0}\right)\},$$

where $\mathrm{d}$ represents the PW’s depth delineated by (10), and (12) is satisfied.

The outer bound of the PW leads to the set:

$$\mathrm{Z}=\left\{\right(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1(\mathsf{\Omega })\times {\mathrm{H}}_0^1(\mathsf{\Omega }\left)\right|\mathrm{E}(\mathrm{u},\mathrm{v})<\mathrm{d},\mathrm{K}(\mathrm{u},\mathrm{v})<0\}.$$

For $\mathsf{\delta }>0$, we further define

$${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})=\mathsf{\delta }{\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}-\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}-2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}$$

(13)

and

$$\mathrm{r}\left(\mathsf{\delta }\right)={\left[{\displaystyle \frac{\mathsf{\delta }}{\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}}}\right]}^{{\displaystyle \frac{1}{\mathrm{p}-1}}}.$$

The subsequent lemmas are presented to denote the relation between $\Vert (\mathrm{u},\mathrm{v})\Vert$ and ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})$.

Lemma 2.

Presuming that $\mathrm{u},\mathrm{v}\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$, if $\Vert (\mathrm{u},\mathrm{v})\Vert <\mathrm{r}\left(\mathrm{\delta }\right)$, then ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})>0$. If $\Vert (\mathrm{u},\mathrm{v})\Vert <\mathrm{r}\left(1\right)$, then $\mathrm{K}(\mathrm{u},\mathrm{v})>0$.

Proof. 

From the definition of ${\mathrm{C}}_{\mathrm{*}}$ and $\Vert (\mathrm{u},\mathrm{v})\Vert <\mathrm{r}\left(\mathsf{\delta }\right)$, we can deduce that

$$\begin{array}{c}\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}\left|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}\right|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\int }_{\mathsf{\Omega }}\left(|\mathrm{u}{|}^{\mathrm{p}+1}+|\mathrm{v}{|}^{\mathrm{p}+1}\right)\mathrm{d}\mathrm{x}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\left(\Vert \mathrm{u}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2+\Vert \mathrm{v}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2\right)}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\Vert (\mathrm{u},\mathrm{v}){\Vert }^{\mathrm{p}+1}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\mathrm{r}{\left(\mathsf{\delta }\right)}^{\mathrm{p}-1}\Vert (\mathrm{u},\mathrm{v})\Vert \hfill \\ =\mathsf{\delta }‖(\mathrm{u},\mathrm{v})‖.\hfill \end{array}$$

Thus, via the definition of ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})$ (13), the lemma is proven. □

Lemma 3.

Presuming that $\mathrm{u},\mathrm{v}\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$, if ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})<0$, then $\Vert (\mathrm{u},\mathrm{v})\Vert >\mathrm{r}\left(\mathrm{\delta }\right)$. If $\mathrm{K}(\mathrm{u},\mathrm{v})<0$, then $\Vert (\mathrm{u},\mathrm{v})\Vert >\mathrm{r}\left(1\right)$.

Proof. 

From ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})<0$ and the definition of ${\mathrm{C}}_{\mathrm{*}}$, we obtain

$$\begin{array}{cc}\hfill \mathsf{\delta }\Vert (\mathrm{u},\mathrm{v}){\Vert }^2& \le \mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\int }_{\mathsf{\Omega }}\left(|\mathrm{u}{|}^{\mathrm{p}+1}+|\mathrm{v}{|}^{\mathrm{p}+1}\right)\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\left(\Vert \mathrm{u}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2+\Vert \mathrm{v}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2\right)}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\Vert (\mathrm{u},\mathrm{v}){\Vert }^{\mathrm{p}+1}\hfill \end{array}$$

which leads to

$$‖\mathrm{u},\mathrm{v}‖>{\left[{\displaystyle \frac{\mathsf{\delta }}{\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}}}\right]}^{{\displaystyle \frac{1}{\mathrm{p}-1}}}=\mathrm{r}\left(\mathsf{\delta }\right).$$

□

Lemma 4.

Presuming that $(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\backslash \left\{\left(0,0\right)\right\}$, if ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})=0$, then $\Vert (\mathrm{u},\mathrm{v})\Vert \ge \mathrm{r}\left(\mathrm{\delta }\right)$. If $\mathrm{K}(\mathrm{u},\mathrm{v})=0$, then $\Vert (\mathrm{u},\mathrm{v})\Vert \ge \mathrm{r}\left(1\right)$.

Proof. 

If ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})=0$ and $(\mathrm{u},\mathrm{v})\ne \left(0,0\right)$, then, from the definition of ${\mathrm{C}}_{\mathrm{*}},$ we can deduce that

$$\begin{array}{cc}\hfill \mathsf{\delta }\Vert (\mathrm{u},\mathrm{v}){\Vert }^2& =\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\int }_{\mathsf{\Omega }}\left(|\mathrm{u}{|}^{\mathrm{p}+1}+|\mathrm{v}{|}^{\mathrm{p}+1}\right)\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\left(\Vert \mathrm{u}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2+\Vert \mathrm{v}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2\right)}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\Vert (\mathrm{u},\mathrm{v}){\Vert }^{\mathrm{p}+1}\hfill \end{array}$$

which leads to

$$‖\mathrm{u},\mathrm{v}‖\ge {\left[{\displaystyle \frac{\mathsf{\delta }}{\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}}}\right]}^{{\displaystyle \frac{1}{\mathrm{p}-1}}}=\mathrm{r}\left(\mathsf{\delta }\right).$$

□

For $\mathsf{\delta }>0$, a family of potential wells’ depth is delineated as

$$\mathrm{d}\left(\mathsf{\delta }\right)={\mathrm{i}\mathrm{n}\mathrm{f}}_{\left(\mathrm{u},\mathrm{v}\right)\in {\mathcal{N}}_{\mathsf{\delta }}}\mathrm{E}\left(\mathrm{u},\mathrm{v}\right)$$

(14)

where the Nehari manifold is

$${\mathcal{N}}_{\mathsf{\delta }}=\left\{\right(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1(\mathsf{\Omega })\times {\mathrm{H}}_0^1(\mathsf{\Omega }\left)\right|{\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})=0,(\mathrm{u},\mathrm{v})\ne \left(\mathrm{0,0}\right)\}.$$

(15)

Thus, $\mathrm{d}\left(\mathsf{\delta }\right)$ with a mathematical expression can be estimated as follows:

Lemma 5.

When $0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$, then

$$\mathrm{d}\left(\mathrm{\delta }\right)\ge \mathrm{a}\left(\mathrm{\delta }\right){\mathrm{r}}^2\left(\mathrm{\delta }\right),$$

where $\mathrm{a}\left(\mathrm{\delta }\right)=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathrm{\delta }}{\mathrm{p}+1}}\right)$. In particular,

$$\mathrm{d}\ge {\displaystyle \frac{\mathrm{p}-1}{2(\mathrm{p}+1)}}{\left[{\displaystyle \frac{1}{(\mathrm{p}+1)(2^{\mathrm{p}}\mathrm{\alpha }+\mathrm{\beta }){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}}}\right]}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}>0.$$

Moreover, we obtain

$$\mathrm{d}\left(\mathrm{\delta }\right)={\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathrm{\delta }}{\mathrm{p}+1}}\right){\mathrm{\delta }}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}\mathrm{d}.$$

(16)

Proof. 

From the definition of ${\mathcal{N}}_{\mathsf{\delta }}$ by $\left(15\right)$, we obtain $\Vert (\mathrm{u},\mathrm{v})\Vert \ge \mathrm{r}\left(\mathsf{\delta }\right)$ for $(\mathrm{u},\mathrm{v})\in {\mathcal{N}}_{\mathsf{\delta }}$ via Lemma 4. Hence, based on the descriptions of $\mathrm{E}(\mathrm{u},\mathrm{v})$ and ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})$ using (6) and (13), we deduce that

$$\begin{array}{cc}\hfill \mathrm{E}(\mathrm{u},\mathrm{v})& =\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right)\left(\Vert \nabla \mathrm{u}{\Vert }_{{\mathrm{L}}_2\left(\mathsf{\Omega }\right)}^2+\Vert \nabla \mathrm{v}{\Vert }_{{\mathrm{L}}_2\left(\mathsf{\Omega }\right)}^2\right)+{\displaystyle \frac{1}{\mathrm{p}+1}}{\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})\hfill \\ & \ge \left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right)\Vert (\mathrm{u},\mathrm{v}){\Vert }^2\ge \mathrm{a}\left(\mathsf{\delta }\right){\mathrm{r}}^2\left(\mathsf{\delta }\right)\hfill \end{array}$$

(17)

for $(\mathrm{u},\mathrm{v})\in {\mathcal{N}}_{\mathsf{\delta }}$, which implies that

$$\mathrm{d}\left(\mathsf{\delta }\right)\ge \mathrm{a}\left(\mathsf{\delta }\right){\mathrm{r}}^2\left(\mathsf{\delta }\right)$$

using the description of $\mathrm{d}\left(\mathsf{\delta }\right)$ once, $0<\mathsf{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$.

The lemma’s initial section was proven. Thus, Equation (16) will be proven as follows:

(1) If $\mathsf{\delta }>0,(\overline{\mathrm{u}},\overline{\mathrm{v}})\in {\mathcal{N}}_{\mathsf{\delta }}$ minimizes $\mathrm{d}\left(\mathsf{\delta }\right)={\mathrm{i}\mathrm{n}\mathrm{f}}_{(\mathrm{u},\mathrm{v})\in {\mathcal{N}}_{\mathsf{\delta }}}\mathrm{E}(\mathrm{u},\mathrm{v})$, i.e., $\mathrm{E}(\overline{\mathrm{u}},\overline{\mathrm{v}})=\mathrm{d}\left(\mathsf{\delta }\right)$, it follows that $\mathsf{\lambda }=\mathsf{\lambda }\left(\mathsf{\delta }\right)$ is defined by $(\mathsf{\lambda }\overline{\mathrm{u}},\mathsf{\lambda }\overline{\mathrm{v}})\in \mathcal{N}$ in this case. Therefore, $\mathsf{\lambda }$ satisfies

$$\mathsf{\lambda }={\left({\displaystyle \frac{{\int }_{\mathsf{\Omega }}\left(|\nabla \overline{\mathrm{u}}{|}^2+|\nabla \overline{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x}}{\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\overline{\mathrm{u}}+\overline{\mathrm{v}}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\overline{\mathrm{u}}\overline{\mathrm{v}}{|}^{\frac{\mathrm{p}+1}{2}}\mathrm{d}\mathrm{x}}}\right)}^{{\displaystyle \frac{1}{\mathrm{p}-1}}}={\mathsf{\delta }}^{-{\displaystyle \frac{1}{\mathrm{p}-1}}}$$

and it exists uniquely for every $\mathsf{\delta }>0$. Hence, based on the description of $\mathrm{d}\left(\mathsf{\delta }\right)$ by (14) for $\mathsf{\delta }=1$ and $(\mathsf{\lambda }\overline{\mathrm{u}},\mathsf{\lambda }\overline{\mathrm{v}})\in \mathcal{N}$,

$$\begin{array}{cc}\hfill \mathrm{d}\le \mathrm{E}(\mathsf{\lambda }\overline{\mathrm{u}},\mathsf{\lambda }\overline{\mathrm{v}})& ={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\mathsf{\lambda }}^2{\int }_{\mathsf{\Omega }}\left(|\nabla \overline{\mathrm{u}}{|}^2+|\nabla \overline{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x}+{\displaystyle \frac{1}{\mathrm{p}+1}}\mathrm{K}(\mathsf{\lambda }\overline{\mathrm{u}},\mathsf{\lambda }\overline{\mathrm{v}})\hfill \\ & ={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\mathsf{\delta }}^{-{\displaystyle \frac{2}{\mathrm{p}-1}}}{\int }_{\mathsf{\Omega }}\left(|\nabla \overline{\mathrm{u}}{|}^2+|\nabla \overline{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x}\hfill \end{array}$$

is attained.

Note that when

$$\mathrm{d}\left(\mathsf{\delta }\right)=\mathrm{E}(\overline{\mathrm{u}},\overline{\mathrm{v}})=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right){\int }_{\mathsf{\Omega }}\left(|\nabla \overline{\mathrm{u}}{|}^2+|\nabla \overline{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x},$$

we obtain

$$\mathrm{d}\le {\mathsf{\delta }}^{-{\displaystyle \frac{2}{\mathrm{p}-1}}}{\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right)}^{-1}\mathrm{d}\left(\mathsf{\delta }\right)$$

which implies that

$$\mathrm{d}\left(\mathsf{\delta }\right)\ge {\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right){\mathsf{\delta }}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}\mathrm{d}$$

(18)

for $0<\mathsf{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$.

(2) If $\mathsf{\delta }>0,(\tilde{\mathrm{u}},\tilde{\mathrm{v}})\in \mathcal{N}$ minimizes $\mathrm{d}={\mathrm{i}\mathrm{n}\mathrm{f}}_{(\mathrm{u},\mathrm{v})\in \mathcal{N}}\mathrm{E}(\mathrm{u},\mathrm{v})$, i.e., $\mathrm{E}(\tilde{\mathrm{u}},\tilde{\mathrm{v}})=\mathrm{d}$, then $\mathsf{\lambda }=\mathsf{\lambda }\left(\mathsf{\delta }\right)$ is delineated by $(\tilde{\mathrm{u}},\tilde{\mathrm{v}})\in {\mathcal{N}}_{\mathsf{\delta }}$. Thus, $\mathsf{\lambda }$ satisfies

$$\mathsf{\lambda }={\left({\displaystyle \frac{\mathsf{\delta }{\int }_{\mathsf{\Omega }}\left(|\nabla \tilde{\mathrm{u}}{|}^2+|\nabla \tilde{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x}}{\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\tilde{\mathrm{u}}+\tilde{\mathrm{v}}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\tilde{\mathrm{u}}\tilde{\mathrm{v}}{|}^{\frac{\mathrm{p}+1}{2}}\mathrm{d}\mathrm{x}}}\right)}^{{\displaystyle \frac{1}{\mathrm{p}-1}}}={\mathsf{\delta }}^{{\displaystyle \frac{1}{\mathrm{p}-1}}}$$

and it exists uniquely for each $\mathsf{\delta }>0$. Thus, from the definition of $\mathrm{d}\left(\mathsf{\delta }\right)$ by (14) and $(\mathsf{\lambda }\tilde{\mathrm{u}},\mathsf{\lambda }\tilde{\mathrm{v}})\in {\mathcal{N}}_{\mathsf{\delta }}$, we can deduce that

$$\begin{array}{cc}\hfill \mathrm{d}\left(\mathsf{\delta }\right)& \le \mathrm{E}(\mathsf{\lambda }\tilde{\mathrm{u}},\mathsf{\lambda }\tilde{\mathrm{v}})=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right){\mathsf{\lambda }}^2{\int }_{\mathsf{\Omega }}\left(|\nabla \tilde{\mathrm{u}}{|}^2+|\nabla \tilde{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x}+{\displaystyle \frac{1}{\mathrm{p}+1}}{\mathrm{K}}_{\mathsf{\delta }}(\mathsf{\lambda }\tilde{\mathrm{u}},\mathsf{\lambda }\tilde{\mathrm{v}})\hfill \\ & =\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right){\mathsf{\delta }}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}{\int }_{\mathsf{\Omega }}\left(|\nabla \tilde{\mathrm{u}}{|}^2+|\nabla \tilde{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x}+{\displaystyle \frac{1}{\mathrm{p}+1}}{\mathrm{K}}_{\mathsf{\delta }}(\mathsf{\lambda }\tilde{\mathrm{u}},\mathsf{\lambda }\tilde{\mathrm{v}}).\hfill \end{array}$$

Note that when

$$\mathrm{d}=\mathrm{E}\left(\tilde{\mathrm{u}},\tilde{\mathrm{v}}\right)={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\int }_{\mathsf{\Omega }}\left(|\nabla \tilde{\mathrm{u}}{|}^2+|\nabla \tilde{\mathrm{v}}{|}^2\right)\mathrm{d}\mathrm{x},$$

we obtain

$$\mathrm{d}\left(\mathsf{\delta }\right)\le {\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right){\mathsf{\delta }}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}\mathrm{d}$$

(19)

for $0<\mathsf{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$. From (18) and (19), (16) is obtained. □

Furthermore, the behavior of $\mathrm{d}\left(\mathsf{\delta }\right)$ changes regarding $\mathsf{\delta }$ in the subsequent lemma.

Lemma 6.

The subsequent features of $\mathrm{d}\left(\mathrm{\delta }\right)$ are satisfied by the following:

(i) 

${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathrm{\delta }\to 0}\mathrm{d}\left(\mathrm{\delta }\right)=0,\mathrm{d}\left({\displaystyle \frac{\mathrm{p}+1}{2}}\right)=0$,  $\mathrm{d}\left(\mathrm{\delta }\right)$ has continuity, and $\mathrm{d}\left(\mathrm{\delta }\right)>0$ for $0<\mathrm{\delta }\le {\displaystyle \frac{\mathrm{p}+1}{2}}$.

(ii) 

$\mathrm{d}\left(\mathrm{\delta }\right)$ increases on $0<\mathrm{\delta }\le 1$, decreases upon $1\le \mathrm{\delta }\le {\displaystyle \frac{\mathrm{p}+1}{2}}$, and reaches the highest $\mathrm{d}=\mathrm{d}\left(1\right)$ at $\mathrm{\delta }=1$.

Proof. 

From Lemma 5, the outcome of (i) is instantly attained as follows:

$${\mathrm{d}}^{\prime }\left(\mathsf{\delta }\right)={\displaystyle \frac{2\left(\mathrm{p}+1\right)}{{\left(\mathrm{p}-1\right)}^2}}{\mathsf{\delta }}^{{\displaystyle \frac{2}{\mathrm{p}-1}}-1}\left(1-\mathsf{\delta }\right)$$

which implies that (ii) is concluded. □

Lemma 7.

Presuming that $0<\mathrm{E}(\mathrm{u},\mathrm{v})<\mathrm{d}$ for some $\mathrm{u},\mathrm{v}\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$, and ${\mathrm{\delta }}_1<{\mathrm{\delta }}_2,$ the equation’s two roots $\mathrm{E}(\mathrm{u},\mathrm{v})=\mathrm{d}\left(\mathrm{\delta }\right)$. Thus, ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})$’s sign does not change for $\mathrm{\delta }\in ({\mathrm{\delta }}_1,{\mathrm{\delta }}_2)$.

Proof. 

$\mathrm{E}(\mathrm{u},\mathrm{v})>0$ implies $(\mathrm{u},\mathrm{v})\ne 0$. If ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})$ ’s sign is changeable to $\mathsf{\delta }\in ({\mathsf{\delta }}_1,{\mathsf{\delta }}_2)$, we can choose $\overline{\mathsf{\delta }}\in ({\mathsf{\delta }}_1,{\mathsf{\delta }}_2)$, satisfying ${\mathrm{K}}_{\overline{\mathsf{\delta }}}(\mathrm{u},\mathrm{v})=0$. Hence, from the description of $\mathrm{d}\left(\mathsf{\delta }\right),\mathrm{E}(\mathrm{u},\mathrm{v})\ge \mathrm{d}\left(\overline{\mathsf{\delta }}\right)$ is attained. Nevertheless, based on Lemma 6, $\mathrm{E}\left(\mathrm{u},\mathrm{v}\right)=\mathrm{d}\left({\mathsf{\delta }}_1\right)=\mathrm{d}\left({\mathsf{\delta }}_2\right)<\mathrm{d}\left(\overline{\mathsf{\delta }}\right),$ which contradicts this. □

After the description of the family of potential wells’ depths $\mathrm{d}\left(\mathsf{\delta }\right)$, the subsequent lemma is presented to show the relationship between ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})$ and $\Vert (\mathrm{u},\mathrm{v})\Vert$, when $\mathrm{E}(\mathrm{u},\mathrm{v})\le \mathrm{d}\left(\mathsf{\delta }\right)$.

Lemma 8.

Let $\mathrm{u},\mathrm{v}\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$ and $0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$. Presume that $\mathrm{E}(\mathrm{u},\mathrm{v})\le \mathrm{d}\left(\mathrm{\delta }\right)$.

(i) 

If ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})>0$, then $0<\Vert (\mathrm{u},\mathrm{v}){\Vert }^2<{\displaystyle \frac{\mathrm{d}\left(\mathrm{\delta }\right)}{\mathrm{a}\left(\mathrm{\delta }\right)}}$. If $\mathrm{K}(\mathrm{u},\mathrm{v})>0$, then $0<\Vert (\mathrm{u},\mathrm{v}){\Vert }^2<{\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\mathrm{d}.$

(ii) 

If $\Vert (\mathrm{u},\mathrm{v}){\Vert }^2>{\displaystyle \frac{\mathrm{d}\left(\mathrm{\delta }\right)}{\mathrm{a}\left(\mathrm{\delta }\right)}}$, then ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})<0$. If $\Vert (\mathrm{u},\mathrm{v}){\Vert }^2>{\displaystyle \frac{2(\mathrm{p}+1)}{\mathrm{p}-1}}\mathrm{d}$, then $\mathrm{K}(\mathrm{u},\mathrm{v})<0$.

(iii) 

If ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})=0$, then $\Vert (\mathrm{u},\mathrm{v}){\Vert }^2\le {\displaystyle \frac{\mathrm{d}\left(\mathrm{\delta }\right)}{\mathrm{a}\left(\mathrm{\delta }\right)}}$. If $\mathrm{K}(\mathrm{u},\mathrm{v})=0$, then $\Vert (\mathrm{u},\mathrm{v}){\Vert }^2\le {\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\mathrm{d}$.

Proof. 

(i) If $\mathrm{E}(\mathrm{u},\mathrm{v})\le \mathrm{d}\left(\mathsf{\delta }\right)$ and ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})>0$, we obtain

$$\mathrm{d}(\mathsf{\delta })\ge \mathrm{E}(\mathrm{u},\mathrm{v})=({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}){\int }_{\mathsf{\Omega }}\left(\right|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2)\mathrm{d}\mathrm{x}+{\displaystyle \frac{1}{\mathrm{p}+1}}{\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})>\mathrm{a}(\mathsf{\delta })\Vert (\mathrm{u},\mathrm{v}){\Vert }^2.$$

(ii) If $\mathrm{E}(\mathrm{u},\mathrm{v})\le \mathrm{d}\left(\mathsf{\delta }\right)$ and $\Vert (\mathrm{u},\mathrm{v}){\Vert }^2>{\displaystyle \frac{\mathrm{d}\left(\mathsf{\delta }\right)}{\mathrm{a}\left(\mathsf{\delta }\right)}}$, then

$$\begin{array}{cc}& {\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})=(\mathrm{p}+1)\mathrm{E}(\mathrm{u},\mathrm{v})-({\displaystyle \frac{\mathrm{p}+1}{2}}-\mathsf{\delta })(\Vert \nabla \mathrm{u}{\Vert }_{{\mathrm{L}}_2(\mathsf{\Omega })}^2+\Vert \nabla \mathrm{v}{\Vert }_{{\mathrm{L}}_2(\mathsf{\Omega })}^2)\\ & <(\mathrm{p}+1)\mathrm{d}(\mathsf{\delta })-({\displaystyle \frac{\mathrm{p}+1}{2}}-\mathsf{\delta }){\displaystyle \frac{\mathrm{d}(\mathsf{\delta })}{\mathrm{a}(\mathsf{\delta })}}=0.\end{array}$$

(iii) If $\mathrm{E}(\mathrm{u},\mathrm{v})\le \mathrm{d}\left(\mathsf{\delta }\right)$ and ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})=0$, then

$$\mathrm{d}\left(\mathsf{\delta }\right)\ge \mathrm{E}(\mathrm{u},\mathrm{v})=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{\mathsf{\delta }}{\mathrm{p}+1}}\right)\left(\Vert \nabla \mathrm{u}{\Vert }_{{\mathrm{L}}_2\left(\mathsf{\Omega }\right)}^2+\Vert \nabla \mathrm{v}{\Vert }_{{\mathrm{L}}_2\left(\mathsf{\Omega }\right)}^2\right)+{\displaystyle \frac{1}{\mathrm{p}+1}}{\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})=\mathrm{a}\left(\mathsf{\delta }\right)\Vert (\mathrm{u},\mathrm{v}){\Vert }^2.$$

□

Remark 2.

The outcomes of Lemmas 2–4 and Lemma 8 designate that ${\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$ is split into two sections, ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})>0$ and ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})<0,$ with the surface ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})=0\left((\mathrm{u},\mathrm{v})\ne (0, 0)\right)$. The inner section of ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})=0$ is ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})>0$ and the outer section of ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})=0$ is ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})<0$. The sphere $\Vert (\mathrm{u},\mathrm{v}){\Vert }^{ }=\mathrm{r}\left(\mathrm{\delta }\right)$ sits inside ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})>0$ and the sphere $\Vert (\mathrm{u},\mathrm{v}){\Vert }^2={\displaystyle \frac{\mathrm{d}\left(\mathrm{\delta }\right)}{\mathrm{a}\left(\mathrm{\delta }\right)}}$ sits inside ${\mathrm{K}}_{\mathrm{\delta }}(\mathrm{u},\mathrm{v})<0$.

A family of PWs is introduced. For $0<\mathsf{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$,

$$\begin{gathered} {\mathrm{W}}_{\mathsf{\delta }}=\left\{\right(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1(\mathsf{\Omega })\times {\mathrm{H}}_0^1(\mathsf{\Omega }\left)\right|{\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})>0,\mathrm{E}(\mathrm{u},\mathrm{v})<\mathrm{d}\left(\mathsf{\delta }\right)\}\cup \{(0,0)\}, \\ {\overline{\mathrm{W}}}_{\mathsf{\delta }}={\mathrm{W}}_{\mathsf{\delta }}\cap \partial {\mathrm{W}}_{\mathsf{\delta }}=\left\{\left(\mathrm{u},\mathrm{v}\right)\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)|{\mathrm{K}}_{\mathsf{\delta }}\left(\mathrm{u},\mathrm{v}\right)\ge 0,\mathrm{E}\left(\mathrm{u},\mathrm{v}\right)<\mathrm{d}\left(\mathsf{\delta }\right)\right\}. \end{gathered}$$

Moreover, the outer parts of the relevant PW sets are delineated as follows:

$${\mathrm{Z}}_{\mathsf{\delta }}=\left\{\right(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1(\mathsf{\Omega })\times {\mathrm{H}}_0^1(\mathsf{\Omega }\left)\right|{\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})<0,\mathrm{E}(\mathrm{u},\mathrm{v})<\mathrm{d}\left(\mathsf{\delta }\right)\}$$

for $0<\mathsf{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$.

Lemma 9.

Suppose that $0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$. Thus,

(i) 

${\mathrm{B}}_{{\mathrm{r}}_1\left(\mathrm{\delta }\right)}\subset {\mathrm{W}}_{\mathrm{\delta }}\subset {\mathrm{B}}_{{\mathrm{r}}_2\left(\mathrm{\delta }\right)}$, where

$$\begin{gathered} { \mathrm{B}}_{{\mathrm{r}}_1\left(\mathrm{\delta }\right)}=\{(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1(\mathrm{\Omega })\times {\mathrm{H}}_0^1(\mathrm{\Omega }\left)\right|\left.‖\left(\mathrm{u},\mathrm{v}\right){\Vert }^2<\mathrm{m}\mathrm{i}\mathrm{n}\left\{{\mathrm{r}}^2\left(\mathrm{\delta }\right),2\mathrm{d}\left(\mathrm{\delta }\right)\right\}\right\}, \\ {\mathrm{B}}_{{\mathrm{r}}_2\left(\mathrm{\delta }\right)}=\left\{\left(\mathrm{u},\mathrm{v}\right)\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)|‖\mathrm{u},\mathrm{v}‖<{\displaystyle \frac{\mathrm{d}\left(\mathrm{\delta }\right)}{\mathrm{a}\left(\mathrm{\delta }\right)}}\right\}. \end{gathered}$$

(ii) 

${\mathrm{Z}}_{\mathrm{\delta }}\subset {\mathrm{B}}_{\mathrm{\delta }}^{\mathrm{c}}$, where

$${ \mathrm{B}}_{\mathrm{\delta }}=\left\{\left(\mathrm{u},\mathrm{v}\right)\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)|‖\mathrm{u},\mathrm{v}‖<\mathrm{r}\left(\mathrm{\delta }\right)\right\}.$$

Proof. 

(i) If $(\mathrm{u},\mathrm{v})\in {\mathrm{B}}_{{\mathrm{r}}_1\left(\mathsf{\delta }\right)}$, then $\Vert (\mathrm{u},\mathrm{v})\Vert <\mathrm{r}\left(\mathsf{\delta }\right) \mathrm{a}\mathrm{n}\mathrm{d} \Vert (\mathrm{u},\mathrm{v}){\Vert }^2<2\mathrm{d}\left(\mathsf{\delta }\right).$

Then, from Lemma 2 and $\mathrm{E}(\mathrm{u},\mathrm{v})\le {\displaystyle \frac{1}{2}}\Vert (\mathrm{u},\mathrm{v}){\Vert }^2$, ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})>0$ and $\mathrm{E}(\mathrm{u},\mathrm{v})<\mathrm{d}\left(\mathsf{\delta }\right)$ are obtained. Hence, we can deduce that $\left(\mathrm{u},\mathrm{v}\right)\in {\mathrm{W}}_{\mathsf{\delta }}$.

If $\left(\mathrm{u},\mathrm{v}\right)\in {\mathrm{W}}_{\mathsf{\delta }}$, from Lemma 8, we can determine that $\Vert (\mathrm{u},\mathrm{v}){\Vert }^2<{\displaystyle \frac{\mathrm{d}\left(\mathsf{\delta }\right)}{\mathrm{a}\left(\mathsf{\delta }\right)}},$ from which we can determine that $(\mathrm{u},\mathrm{v})\in {\mathrm{B}}_{{\mathrm{r}}_2\left(\mathsf{\delta }\right)}$.

(ii) If $(\mathrm{u},\mathrm{v})\in {\mathrm{Z}}_{\mathsf{\delta }}$, from Lemma 3, we can deduce that $‖\mathrm{u},\mathrm{v}‖>\mathrm{r}\left(\mathsf{\delta }\right),$ from which we can obtain $(\mathrm{u},\mathrm{v})\in {\mathrm{B}}_{\mathsf{\delta }}^{\mathrm{c}}$. □

Based on the definitions of ${\mathrm{W}}_{\mathsf{\delta }}$ and ${\mathrm{Z}}_{\mathsf{\delta }}$ and Lemma 6:

Lemma 10.

The PW sets and their outer parts satisfy the subsequent features:

(i) 

If $0<{\mathrm{\delta }}^{\prime }<{\mathrm{\delta }}^{″}\le 1$, then ${\mathrm{W}}_{{\mathrm{\delta }}^{\prime }}\subset {\mathrm{W}}_{{\mathrm{\delta }}^{″}}$.

(ii) 

If $1\le {\mathrm{\delta }}^{\prime }<{\mathrm{\delta }}^{″}<{\displaystyle \frac{\mathrm{p}+1}{2}}$, then ${\mathrm{Z}}_{{\mathrm{\delta }}^{\prime }}\subset {\mathrm{Z}}_{{\mathrm{\delta }}^{″}}$.

2.2. The Solution’s Properties of Invariance Sets and Vacuum Isolation

Here, we discuss some sets’ invariance properties of Equation (1)’s flow and the behavior of vacuum isolation, which will be used in the following sections.

Proposition 1.

Assume that ${\mathrm{u}}_0,{\mathrm{v}}_0\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$. Let $0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$. Presume that $0<\mathrm{l}<\mathrm{d},{ \mathrm{\delta }}_1<{\mathrm{\delta }}_2$denotes the equation’s two roots $\mathrm{d}\left(\mathrm{\delta }\right)=\mathrm{l}$. Thus, all solutions to Equation (1) on $0<\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{l},\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$ belong to ${\mathrm{W}}_{\mathrm{\delta }}$ for ${\mathrm{\delta }}_1<\mathrm{\delta }<{\mathrm{\delta }}_2$.

Proof. 

Let $\left(\mathrm{u}\left(\mathrm{t}\right),\mathrm{v}\left(\mathrm{t}\right)\right)$ be any solution to Equation (1) for $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{l},\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)>0,\mathrm{T}$, which denotes the highest existence duration of $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)$. From Lemma 6, it follows that

$$0<\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{l}=\mathrm{d}\left({\mathsf{\delta }}_1\right)=\mathrm{d}\left({\mathsf{\delta }}_2\right)<\mathrm{d}\left(\mathsf{\delta }\right)<\mathrm{d}$$

for ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$. From Lemma 7, it follows that ${\mathrm{K}}_{\mathsf{\delta }}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$ for ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$.

Hence, we can deduce that $\left({\mathrm{u}}_0,{\mathrm{v}}_0\right)\in {\mathrm{W}}_{\mathsf{\delta }}$ for ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$.

Our goal is to prove that $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\in {\mathrm{W}}_{\mathsf{\delta }}$ for $\mathrm{t}\in [0,\mathrm{T})$ and ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$ by contradiction. Thus, we presume that ${\mathsf{\delta }}_0\in ({\mathsf{\delta }}_1,{\mathsf{\delta }}_2)$ and ${\mathrm{t}}_0\in (0,\mathrm{T})$ exist such that $\left(\mathrm{u}\left({\mathrm{t}}_0\right),\mathrm{v}\left({\mathrm{t}}_0\right)\right)\notin {\mathrm{W}}_{{\mathsf{\delta }}_0},$ which means that $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ge \mathrm{d}\left({\mathsf{\delta }}_0\right)$, or ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\le 0$ and $\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ne \left(\mathrm{0,0}\right)$.

If ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)=0$ and $\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ne \left(\mathrm{0,0}\right)$, then $\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\in {\mathcal{N}}_{{\mathsf{\delta }}_0}$ based on the description of ${\mathcal{N}}_{\mathsf{\delta }}$. From the description of $\mathrm{d}\left(\mathsf{\delta }\right)$, $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ge \mathrm{d}\left({\mathsf{\delta }}_0\right)$ is obtained.

If ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)<0$ and $\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ne \left(\mathrm{0,0}\right)$, based on the time continuity of ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})$ and ${\mathrm{K}}_{{\mathsf{\delta }}_0}({\mathrm{u}}_0,{\mathrm{v}}_0)>0,$ we can determine that there exists at least one $\mathrm{s}\in (0,{\mathrm{t}}_0)$ such that ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{s}),\mathrm{v}(\mathrm{s}\left)\right)=0$:

$${\mathrm{t}}^{\mathrm{*}}:=\mathrm{s}\mathrm{u}\mathrm{p}\left\{\mathrm{s}\in (0,{\mathrm{t}}_0)\mid {\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{s}),\mathrm{v}(\mathrm{s}\left)\right)=0\right\}.$$

Consequently, ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)=0$ and ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<0$ for $\mathrm{t}\in ({\mathrm{t}}^{\mathrm{*}},{\mathrm{t}}_0)$.

Two cases are considered, as follows:

Initial case: $\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)\ne \left(0,0\right)$. Then, $\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)\in {\mathcal{N}}_{{\mathsf{\delta }}_0}$, based on the description of ${\mathcal{N}}_{\mathsf{\delta }}$. From the description of $\mathrm{d}\left(\mathsf{\delta }\right)$, $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)\ge \mathrm{d}\left({\mathsf{\delta }}_0\right)$ is obtained. Due to the energy conservation (7),

$$\mathrm{E}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\le \mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}\left(\mathsf{\delta }\right)$$

is noted for $\mathrm{t}\in \left[0,\mathrm{T}\right)$ and ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$. Hence, $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ge \mathrm{d}\left(\mathsf{\delta }\right)$ is impossible for any ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$.

Second case: $\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)=\left(\mathrm{0,0}\right)$. In this case, it is necessary that ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<0$ for $\mathrm{t}\in ({\mathrm{t}}^{\mathrm{*}},{\mathrm{t}}_0)$ and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathrm{t}\to {\mathrm{t}}^{\mathrm{*}+}}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)=0$. Thus, from Lemma 3, $\Vert \left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\Vert >\mathrm{r}\left({\mathsf{\delta }}_0\right)$ for $\mathrm{t}\in ({\mathrm{t}}^{\mathrm{*}},{\mathrm{t}}_0)$ is attained, which contradicts with ${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathrm{t}\to {\mathrm{t}}^{\mathrm{*}}}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)=0$. □

Proposition 2.

Assume that ${\mathrm{u}}_0,{\mathrm{v}}_0\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$. Let $0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$. Presume that $0<\mathrm{l}<\mathrm{d},{ \mathrm{\delta }}_1<{\mathrm{\delta }}_2 \mathrm{f}\mathrm{o}\mathrm{r}$ the equation’s two roots $\mathrm{d}\left(\mathrm{\delta }\right)=\mathrm{l}$. Thus, for all solutions to Equation (1), $0<\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{l}, \mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)<0$ belongs to ${\mathrm{Z}}_{\mathrm{\delta }}$ for ${\mathrm{\delta }}_1<\mathrm{\delta }<{\mathrm{\delta }}_2$.

Proof. 

Let $\left(\mathrm{u}\left(\mathrm{t}\right),\mathrm{v}\left(\mathrm{t}\right)\right)$ be any solution to Equation (1) for $\mathrm{E}\left({\mathrm{u}}_0,{\mathrm{v}}_0\right)=\mathrm{l},\mathrm{K}\left({\mathrm{u}}_0,{\mathrm{v}}_0\right)<0,\mathrm{T},\mathrm{w}\mathrm{h}\mathrm{i}\mathrm{c}\mathrm{h}$ denotes the highest existence duration of $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)$. From Lemma 6, it follows that

$$0<\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{l}=\mathrm{d}\left({\mathsf{\delta }}_1\right)=\mathrm{d}\left({\mathsf{\delta }}_2\right)<\mathrm{d}\left(\mathsf{\delta }\right)<\mathrm{d}$$

for ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$. From Lemma 7, it follows that for ${\mathrm{K}}_{\mathsf{\delta }}({\mathrm{u}}_0,{\mathrm{v}}_0)<0$ and ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$.

Hence, we can deduce that $({\mathrm{u}}_0,{\mathrm{v}}_0)\in {\mathrm{Z}}_{\mathsf{\delta }}$ for ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$.

Our goal is to prove that $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\in {\mathrm{Z}}_{\mathsf{\delta }}$ for $\mathrm{t}\in [0,\mathrm{T})$ and ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$ by contradiction. Thus, presume that ${\mathsf{\delta }}_0\in ({\mathsf{\delta }}_1,{\mathsf{\delta }}_2)$ and ${\mathrm{t}}_0\in (0,\mathrm{T})$, which means that

$$\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ge \mathrm{d}\left({\mathsf{\delta }}_0\right) \mathrm{or} {\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ge 0$$

If ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ge 0$, based on the time continuity of ${\mathrm{K}}_{\mathsf{\delta }}(\mathrm{u},\mathrm{v})$ and ${\mathrm{K}}_{{\mathsf{\delta }}_0}({\mathrm{u}}_0,{\mathrm{v}}_0)<0,$ we can determine that there exists at least one $\mathrm{s}\in (0,{\mathrm{t}}_0)$ such that ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{s}),\mathrm{v}(\mathrm{s}\left)\right)=0$:

$${\mathrm{t}}^{\mathrm{*}}:=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\mathrm{s}\in (0,{\mathrm{t}}_0)\mid {\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{s}),\mathrm{v}(\mathrm{s}\left)\right)=0\right\}.$$

Consequently, ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)=0$ and ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<0$ for $\mathrm{t}\in (0,{\mathrm{t}}^{\mathrm{*}})$.

Two cases are considered.

Initial case: $\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)\ne (0,0)$. Then, $\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)\in {\mathcal{N}}_{{\mathsf{\delta }}_0}$ based on the description of ${\mathcal{N}}_{\mathsf{\delta }}$. From the description of $\mathrm{d}\left(\mathsf{\delta }\right)$, $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)\ge \mathrm{d}\left({\mathsf{\delta }}_0\right)$ is obtained. Due to the energy conservation of (7), $\mathrm{E}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\le \mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}\left(\mathsf{\delta }\right)$ is noted for $\mathrm{t}\in \left[0,\mathrm{T}\right)$ and ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$. Hence, $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)\ge \mathrm{d}\left(\mathsf{\delta }\right)$ is impossible for any ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2$.

Second case: $\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)=(0,0)$. In this case, we must have ${\mathrm{K}}_{{\mathsf{\delta }}_0}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<0$ for $\mathrm{t}\in (0,{\mathrm{t}}^{\mathrm{*}})$ and ${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathrm{t}\to {\mathrm{t}}^{\mathrm{*}}}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)=0$. Thus, from Lemma 3, $\Vert \left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\Vert >\mathrm{r}\left({\mathsf{\delta }}_0\right)$ for $\mathrm{t}\in (0,{\mathrm{t}}^{\mathrm{*}})$ is obtained, which contradicts with ${\mathrm{l}\mathrm{i}\mathrm{m}}_{\mathrm{t}\to {\mathrm{t}}^{\mathrm{*}-}}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)=0$. □

The previously expressed propositions indicate the invariance properties of ${\mathrm{W}}_{\mathsf{\delta }}$ and ${\mathrm{Z}}_{\mathsf{\delta }}$. Furthermore, when their conjunctions are a concern regarding $\mathsf{\delta }$, the following proposition can be obtained.

Proposition 3.

Assume that ${\mathrm{u}}_0,{\mathrm{v}}_0\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$. Let $0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$ be presented.

Presume that $0<\mathrm{l}<\mathrm{d},{ \mathrm{\delta }}_1<{\mathrm{\delta }}_2$ denote the equation’s two roots $\mathrm{d}\left(\mathrm{\delta }\right)=\mathrm{l}$. Thus, for $\mathrm{\delta }\in ({\mathrm{\delta }}_1,{\mathrm{\delta }}_2)$ both sets

$${\mathrm{W}}_{{\mathrm{\delta }}_1{\mathrm{\delta }}_2}={\cup }_{{\mathrm{\delta }}_1<\mathrm{\delta }<{\mathrm{\delta }}_2} {\mathrm{W}}_{\mathrm{\delta }} \mathrm{a}\mathrm{n}\mathrm{d}{ \mathrm{Z}}_{{\mathrm{\delta }}_1{\mathrm{\delta }}_2}={\cup }_{{\mathrm{\delta }}_1<\mathrm{\delta }<{\mathrm{\delta }}_2} {\mathrm{Z}}_{\mathrm{\delta }}$$

have invariance properties. Equation (1)’ s flow is a concern if $0<\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{e}$.

The subsequent proposition claims that, between these two invariant manifolds ${\mathrm{W}}_{{\mathsf{\delta }}_1{\mathsf{\delta }}_2}$ and ${\mathrm{Z}}_{{\mathsf{\delta }}_1{\mathsf{\delta }}_2}$, a vacuum area exists, where any solutions are available.

Proposition 4.

Assume that ${\mathrm{u}}_0,{\mathrm{v}}_0\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$. Let $0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}$. Presume that $0<\mathrm{l}<\mathrm{d}, {\mathrm{\delta }}_1<{\mathrm{\delta }}_2$ denote the equation’s two roots $\mathrm{d}\left(\mathrm{\delta }\right)=\mathrm{l}$. Thus, for all solutions of Equation (1) with $0<\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{l}$,$(\mathrm{u},\mathrm{v})\notin {\mathrm{W}}_{{\mathrm{\delta }}_1{\mathrm{\delta }}_2}={\cup }_{{\mathrm{\delta }}_1<\mathrm{\delta }<{\mathrm{\delta }}_2}, {\mathrm{W}}_{\mathrm{\delta }}$ is obtained.

Remark 3.

The vacuum area becomes larger and larger as $\mathrm{l}$ increases and

$${\mathcal{N}}_0={\cup }_{0<\mathrm{\delta }<{\displaystyle \frac{\mathrm{p}+1}{2}}} {\mathcal{N}}_{\mathrm{\delta }}$$

is attained as the limited situation.

3. Main Results

The proofs of Theorems 1 and 2 are presented in Section 3.1 and Section 3.2, respectively.

3.1. Global Existence with the Exponential Decay of the Solution

Komornik’s subsequent lemma is introduced and plays a pivotal role when Equation (1)’s global solution, with exponentially decayed features, is examined [20].

Lemma 11.

Let $\mathrm{y}\left(\mathrm{t}\right):{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ denote a non-increasing mapping, and presume that $\mathrm{A}>0$ is a fixed value, such that

$${\int }_{\mathrm{s}}^{+\infty } \mathrm{y}\left(\mathrm{t}\right)\mathrm{d}\mathrm{t}\le \mathrm{A}\mathrm{y}\left(\mathrm{s}\right), 0\le \mathrm{s}<+\infty$$

Then, for all $\mathrm{t}\ge 0$ we obtain

$$\mathrm{y}\left(\mathrm{t}\right)\le \mathrm{y}\left(0\right){\mathrm{e}}^{1-{\displaystyle \frac{\mathrm{t}}{\mathrm{A}}}}.$$

Theorem 1 is thus proven.

Proof of Theorem 1.

Three steps are used for this proof.

Step 1. Proof when there is initially global existence for a low-energy case. When $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}$, based on the description of $\mathrm{d}$, $\mathrm{K}\left({\mathrm{u}}_0,{\mathrm{v}}_0\right)=0$ does not exist. Thus: $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}$ and $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$ need only be considered.

Let $\left\{{\mathsf{\phi }}_{\mathrm{j}}\right(\mathrm{x}\left)\right\}$ denote the base function’s system in ${\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$. The approximate solution to Equation (1) is constructed as follows:

$$\begin{gathered} {\mathrm{u}}_{\mathrm{m}}(\mathrm{t},\mathrm{x})={\sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{g}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{t}\right){\mathsf{\phi }}_{\mathrm{j}}\left(\mathrm{x}\right),\mathrm{m}=\mathrm{1,2},\cdots \\ {\mathrm{v}}_{\mathrm{m}}(\mathrm{t},\mathrm{x})={\sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{h}}_{\mathrm{j}\mathrm{m}}\left(\mathrm{t}\right){\mathsf{\phi }}_{\mathrm{j}}\left(\mathrm{x}\right),\mathrm{m}=\mathrm{1,2},\cdots \end{gathered}$$

(20)

satisfying

$$\begin{gathered} {({\mathrm{u}}_{\mathrm{m}\mathrm{t}},{\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla {\mathrm{u}}_{\mathrm{m}\mathrm{t}},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla {\mathrm{u}}_{\mathrm{m}},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2={\left({\mathrm{f}}_1\right({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}}),{\mathsf{\phi }}_{\mathrm{k}})}_2, \\ {({\mathrm{v}}_{\mathrm{m}\mathrm{t}},{\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla {\mathrm{v}}_{\mathrm{m}\mathrm{t}},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla {\mathrm{v}}_{\mathrm{m}},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2={\left({\mathrm{f}}_2\right({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}}),{\mathsf{\phi }}_{\mathrm{k}})}_2, \end{gathered}$$

(21)

for $\mathrm{k}=\mathrm{1,2},\cdots ,\mathrm{m}$ and as $\mathrm{m}\to +\mathrm{\infty }$,

$$\begin{gathered} {\mathrm{u}}_{\mathrm{m}}(0,\mathrm{x})={\sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{g}}_{\mathrm{j}\mathrm{m}}\left(0\right){\mathsf{\phi }}_{\mathrm{j}}\left(\mathrm{x}\right)\to {\mathrm{u}}_0\left(\mathrm{x}\right), \mathrm{i}\mathrm{n} {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right), \\ {\mathrm{v}}_{\mathrm{m}}\left(0,\mathrm{x}\right)={\sum }_{\mathrm{j}=1}^{\mathrm{m}}{\mathrm{h}}_{\mathrm{j}\mathrm{m}}\left(0\right){\mathsf{\phi }}_{\mathrm{j}}\left(\mathrm{x}\right)\to {\mathrm{v}}_0\left(\mathrm{x}\right), \mathrm{i}\mathrm{n} {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right). \end{gathered}$$

(22)

Multiplying (21) by ${\mathrm{g}}_{\mathrm{k}\mathrm{m}}^{\prime }\left(\mathrm{t}\right)$ and ${\mathrm{h}}_{\mathrm{k}\mathrm{m}}^{\prime }\left(\mathrm{t}\right)$ and then summing for $\mathrm{k},$ we have

$$\begin{gathered} {\left({\mathrm{u}}_{\mathrm{m}}\right(\mathrm{t}),{\mathrm{u}}_{\mathrm{m}}(\mathrm{t}\left)\right)}_2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{(\nabla {\mathrm{u}}_{\mathrm{m}\mathrm{t}},\nabla {\mathrm{u}}_{\mathrm{m}\mathrm{t}})}_2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{(\nabla {\mathrm{u}}_{\mathrm{m}},\nabla {\mathrm{u}}_{\mathrm{m}})}_2 \\ ={\displaystyle \frac{1}{\mathrm{p}+1}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{\left({\mathrm{f}}_1\right({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}}),{\mathrm{u}}_{\mathrm{m}})}_2, \\ {\left({\mathrm{v}}_{\mathrm{m}}\right(\mathrm{t}),{\mathrm{v}}_{\mathrm{m}}(\mathrm{t}\left)\right)}_2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{(\nabla {\mathrm{v}}_{\mathrm{m}\mathrm{t}},\nabla {\mathrm{v}}_{\mathrm{m}\mathrm{t}})}_2+{\displaystyle \frac{1}{2}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{(\nabla {\mathrm{v}}_{\mathrm{m}},\nabla {\mathrm{v}}_{\mathrm{m}})}_2 \\ ={\displaystyle \frac{1}{\mathrm{p}+1}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}{\left({\mathrm{f}}_2\right({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}}),{\mathrm{v}}_{\mathrm{m}})}_2. \end{gathered}$$

(23)

Using (22), we can obtain $\mathrm{E}\left({\mathrm{u}}_{\mathrm{m}}\right(0),{\mathrm{v}}_{\mathrm{m}}(0\left)\right)\to \mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}$ as $\mathrm{m}\to \mathrm{\infty }$. Thus, (23) is integrated in terms of $\mathrm{t}$, and

$${\int }_0^{\mathrm{t}}{\Vert ({\partial }_{\mathsf{\tau }}{\mathrm{u}}_{\mathrm{m}},{\partial }_{\mathsf{\tau }}{\mathrm{v}}_{\mathrm{m}})\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }+\mathrm{E}({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})=\mathrm{E}({\mathrm{u}}_{\mathrm{m}}(0),{\mathrm{v}}_{\mathrm{m}}(0))<\mathrm{d},\hspace{1em}0\le \mathrm{t}<\mathrm{\infty }$$

(24)

is attained when $\mathrm{m}$ is sufficiently large.

Next, we prove that $\left({\mathrm{u}}_{\mathrm{m}}\right(\mathrm{t}),{\mathrm{v}}_{\mathrm{m}}(\mathrm{t}\left)\right)\in \mathrm{W}$ for $\mathrm{m}$ is sufficiently large and $0\le \mathrm{t}<\mathrm{\infty }$. When this is not true, $\mathrm{K}(\mathrm{u},\mathrm{v})$ is continuous, and ${\mathrm{t}}_0>0$ exists such that $\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)\in \partial \mathrm{W}$. Then, $\mathrm{E}\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)=\mathrm{d}$ or $\mathrm{K}\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)=0$ and $\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)\ne (0,0)$. From (24), we can determine that $\mathrm{E}\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)\ne \mathrm{d}$. If $\mathrm{K}\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)=0$ and $\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)\ne$$(0,0)$, then $\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)\in \mathcal{N}$. Based on the following description:

$$\mathrm{d}=\mathrm{i}\mathrm{n}\mathrm{f}\left\{\mathrm{E}(\mathrm{u},\mathrm{v});(\mathrm{u},\mathrm{v})\in \mathcal{N}\right\},$$

$\mathrm{E}\left({\mathrm{u}}_{\mathrm{m}}\right({\mathrm{t}}_0),{\mathrm{v}}_{\mathrm{m}}({\mathrm{t}}_0\left)\right)\ge \mathrm{d}$ is obtained, which contradicts (24).

Hence, from (24) and

$$\begin{array}{cc}\hfill \mathrm{E}({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})& ={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}\left(\Vert \nabla {\mathrm{u}}_{\mathrm{m}}{\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2+\Vert \nabla {\mathrm{v}}_{\mathrm{m}}{\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2\right)+{\displaystyle \frac{1}{\mathrm{p}+1}}\mathrm{K}({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})\hfill \\ & \ge {\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}\Vert \left({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}}\right){\Vert }^2,\hfill \end{array}$$

it follows that

$${\int }_0^{\mathrm{t}}{\Vert ({\partial }_{\mathsf{\tau }}{\mathrm{u}}_{\mathrm{m}},{\partial }_{\mathsf{\tau }}{\mathrm{v}}_{\mathrm{m}})\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }+{\displaystyle \frac{\mathrm{p}-1}{2(\mathrm{p}+1)}}{\Vert ({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})\Vert }^2<\mathrm{d}$$

for $0\le \mathrm{t}<\mathrm{\infty }$ and sufficiently large $\mathrm{m},$ which yields

$$\begin{gathered} {\Vert \nabla {\mathrm{u}}_{\mathrm{m}}\left(\mathrm{t}\right)\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2<{\displaystyle \frac{2(\mathrm{p}+1)}{\mathrm{p}-1}}\mathrm{d},0\le \mathrm{t}<\mathrm{\infty }, \\ {\Vert \nabla {\mathrm{v}}_{\mathrm{m}}\left(\mathrm{t}\right)\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2<{\displaystyle \frac{2(\mathrm{p}+1)}{\mathrm{p}-1}}\mathrm{d},0\le \mathrm{t}<\mathrm{\infty }, \\ {\int }_0^{\mathrm{t}}{‖{\partial }_{\mathsf{\tau }}{\mathrm{u}}_{\mathrm{m}}‖}_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }<\mathrm{d},0\le \mathrm{t}<\mathrm{\infty }, \\ {\int }_0^{\mathrm{t}}{\Vert {\partial }_{\mathsf{\tau }}{\mathrm{v}}_{\mathrm{m}}\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }<\mathrm{d},0\le \mathrm{t}<\mathrm{\infty }. \end{gathered}$$

Based on the Ponicaré inequality, we can obtain the following:

$$\begin{gathered} {\Vert {\mathrm{u}}_{\mathrm{m}}\left(\mathrm{t}\right)\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2\le \mathrm{C}{\Vert \nabla {\mathrm{u}}_{\mathrm{m}}\left(\mathrm{t}\right)\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2<\mathrm{C}{\displaystyle \frac{2(\mathrm{p}+1)}{\mathrm{p}-1}}\mathrm{d},0\le \mathrm{t}<\mathrm{\infty }, \\ {\Vert {\mathrm{v}}_{\mathrm{m}}\left(\mathrm{t}\right)\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2\le \mathrm{C}{\Vert \nabla {\mathrm{v}}_{\mathrm{m}}\left(\mathrm{t}\right)\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2<\mathrm{C}{\displaystyle \frac{2(\mathrm{p}+1)}{\mathrm{p}-1}}\mathrm{d},0\le \mathrm{t}<\mathrm{\infty }. \end{gathered}$$

It follows that $\mathrm{u},\mathrm{v}$ and the subsequences $\left\{{\mathrm{u}}_{\mathsf{\nu }}\right\}$ and $\left\{{\mathrm{v}}_{\mathsf{\nu }}\right\}$ of $\left\{{\mathrm{u}}_{\mathrm{m}}\right\}$ and $\left\{{\mathrm{v}}_{\mathrm{m}}\right\}$ exist such that, as $\mathsf{\nu }\to \mathrm{\infty }$,

${\mathrm{u}}_{\mathsf{\nu }}\to \mathrm{u}$ in ${\mathrm{L}}^{\mathrm{\infty }}(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ weakly star and a.e. in $\mathsf{\Omega }\times [0,\mathrm{\infty })$,

${\mathrm{v}}_{\mathsf{\nu }}\to \mathrm{v}$ in ${\mathrm{L}}^{\mathrm{\infty }}(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ weakly star and a.e. in $\mathsf{\Omega }\times [0,\mathrm{\infty })$,

${\partial }_{\mathrm{t}}{\mathrm{u}}_{\mathsf{\nu }}\to {\partial }_{\mathrm{t}}\mathrm{u}$ in ${\mathrm{L}}^2(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ weakly,

${\partial }_{\mathrm{t}}{\mathrm{v}}_{\mathsf{\nu }}\to {\partial }_{\mathrm{t}}\mathrm{v}$ in ${\mathrm{L}}^2(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ weakly.

Based on the continuity of ${\mathrm{f}}_{\mathrm{i}}\left(\mathrm{u},\mathrm{v}\right),\mathrm{i}=\mathrm{1,2}$, in (21) for constant $\mathrm{k}$ as $\mathrm{m}=\mathsf{\nu }\to \mathrm{\infty }$

$$\begin{array}{cc}{({\mathrm{u}}_{\mathrm{t}},{\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla {\mathrm{u}}_{\mathrm{t}},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla \mathrm{u},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2& ={\left({\mathrm{f}}_1\right(\mathrm{u},\mathrm{v}),{\mathsf{\phi }}_{\mathrm{k}})}_2,\\ {({\mathrm{v}}_{\mathrm{t}},{\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla {\mathrm{v}}_{\mathrm{t}},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2+{(\nabla \mathrm{v},\nabla {\mathsf{\phi }}_{\mathrm{k}})}_2& ={\left({\mathrm{f}}_2\right(\mathrm{u},\mathrm{v}),{\mathsf{\phi }}_{\mathrm{k}})}_2,\end{array}$$

is attained for $\mathrm{k}=\mathrm{1,2},\cdots ,\mathrm{m}$ and

$$\begin{array}{cc}{({\mathrm{u}}_{\mathrm{t}},\mathrm{w})}_2+{(\nabla {\mathrm{u}}_{\mathrm{t}},\nabla \mathrm{w})}_2+{(\nabla \mathrm{u},\nabla \mathrm{w})}_2& ={\left({\mathrm{f}}_1\right(\mathrm{u},\mathrm{v}),\mathrm{w})}_2,\\ {({\mathrm{v}}_{\mathrm{t}},\mathrm{w})}_2+{(\nabla {\mathrm{v}}_{\mathrm{t}},\nabla \mathrm{w})}_2+{(\nabla \mathrm{v},\nabla \mathrm{w})}_2& ={\left({\mathrm{f}}_2\right(\mathrm{u},\mathrm{v}),\mathrm{w})}_2,\end{array}$$

for any $\mathrm{w}\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$ and $\mathrm{t}\in (0,\mathrm{\infty })$. Alternatively, Equation (29) and the WS’s continuous time $\mathrm{u}(\mathrm{x},0)={\mathrm{u}}_0\left(\mathrm{x}\right)$ and $\mathrm{v}(\mathrm{x},0)={\mathrm{v}}_0\left(\mathrm{x}\right)$ for $\mathrm{x}\in \mathsf{\Omega }$ are obtained.

According to the density, we can determine that $\mathrm{u},\mathrm{v}\in {\mathrm{L}}^{\mathrm{\infty }}\left(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\right)$ with ${\mathrm{u}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}\in {\mathrm{L}}^2(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ gives Equation (1)’s global weak solution.

Step 2. Proof when there is global existence for a case of initially overcritical energy. Initially, $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{d}$ implies that $({\mathrm{u}}_0,{\mathrm{v}}_0)\ne 0$. A sequence $\left\{{\mathsf{\lambda }}_{\mathrm{m}}\right\}$ is selected such that $0<{\mathsf{\lambda }}_{\mathrm{m}}<1$, $\mathrm{m}=\mathrm{1,2},\cdots$ and ${\mathsf{\lambda }}_{\mathrm{m}}\to 1$ as $\mathrm{m}\to \mathrm{\infty }$.

Let ${\mathrm{u}}_{0\mathrm{m}}={\mathsf{\lambda }}_{\mathrm{m}}{\mathrm{u}}_0$ and ${\mathrm{v}}_{0\mathrm{m}}={\mathsf{\lambda }}_{\mathrm{m}}{\mathrm{v}}_0$ for $\mathrm{m}=\mathrm{1,2},\cdots$.

Consider the initial conditions following the problem:

$$\left\{\begin{array}{cc}{\partial }_{\mathrm{t}}\mathrm{u}-\mathsf{\Delta }\mathrm{u}-\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}}={\mathrm{F}}_{\mathrm{u}}(\mathrm{u},\mathrm{v}),& \mathrm{i}\mathrm{n}\hspace{1em}\mathsf{\Omega }\times (0,\mathrm{\infty }),\\ {\partial }_{\mathrm{t}}\mathrm{v}-\mathsf{\Delta }\mathrm{v}-\mathsf{\Delta }{\mathrm{v}}_{\mathrm{t}}={\mathrm{F}}_{\mathrm{v}}(\mathrm{u},\mathrm{v}),& \mathrm{i}\mathrm{n}\hspace{1em}\mathsf{\Omega }\times (0,\mathrm{\infty }),\\ \mathrm{u}=0,\mathrm{v}=0,& \mathrm{o}\mathrm{n}\hspace{1em}\mathsf{\Gamma }\times \left(0,\mathrm{\infty }\right),\\ \mathrm{u}(\mathrm{x},0)={\mathrm{u}}_{0\mathrm{m}},\mathrm{v}(\mathrm{x},0)={\mathrm{v}}_{0\mathrm{m}},& \mathrm{x}\in \mathsf{\Omega }.\end{array}\right.$$

(25)

Based on $\mathrm{K}\left({\mathrm{u}}_0,{\mathrm{v}}_0\right)\ge 0$ and Lemma 1, ${\mathsf{\lambda }}^{\mathrm{*}}={\mathsf{\lambda }}^{\mathrm{*}}\left({\mathrm{u}}_0,{\mathrm{v}}_0\right)\ge 1$, $\mathrm{K}\left({\mathrm{u}}_{0\mathrm{m}},{\mathrm{v}}_{0\mathrm{m}}\right)=\mathrm{K}({\mathsf{\lambda }}_{\mathrm{m}}{\mathrm{u}}_0,{\mathsf{\lambda }}_{\mathrm{m}}{\mathrm{v}}_0)>$ 0 and $\mathrm{E}({\mathrm{u}}_{0\mathrm{m}},{\mathrm{v}}_{0\mathrm{m}})=\mathrm{E}({\mathsf{\lambda }}_{\mathrm{m}}{\mathrm{u}}_0,{\mathsf{\lambda }}_{\mathrm{m}}{\mathrm{v}}_0)<\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{d}$ are obtained.

Thus, for $\mathrm{m}\in {\mathbb{N}}_{+}$, it follows from Step 1 and Proposition 1 that a global WS is attained as follows: ${\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}}\in {\mathrm{L}}^{\mathrm{\infty }}(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ with ${\mathrm{u}}_{\mathrm{m}\mathrm{t}},{\mathrm{v}}_{\mathrm{m}\mathrm{t}}\in {\mathrm{L}}^2(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ and $\left({\mathrm{u}}_{\mathrm{m}}\right(\mathrm{t}),{\mathrm{v}}_{\mathrm{m}}(\mathrm{t}\left)\right)\in \mathrm{W}$ for $0\le \mathrm{t}<\mathrm{\infty }$. Therefore, we can determine that

$${\int }_0^{\mathrm{t}}{\Vert ({\partial }_{\mathsf{\tau }}{\mathrm{u}}_{\mathrm{m}},{\partial }_{\mathsf{\tau }}{\mathrm{v}}_{\mathrm{m}})\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }+\mathrm{E}({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})=\mathrm{E}({\mathrm{u}}_{0\mathrm{m}},{\mathrm{v}}_{0\mathrm{m}})<\mathrm{d},\hspace{1em}0\le \mathrm{t}<\mathrm{\infty }.$$

(26)

From (26), $\mathrm{K}\left({\mathrm{u}}_{\mathrm{m}}\right(\mathrm{t}),{\mathrm{v}}_{\mathrm{m}}(\mathrm{t}\left)\right)>0$ for $0\le \mathrm{t}<\mathrm{\infty }$, which means that

$$\begin{array}{cc}\hfill \mathrm{E}({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})& ={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}\left(\Vert \nabla {\mathrm{u}}_{\mathrm{m}}{\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2+\Vert \nabla {\mathrm{v}}_{\mathrm{m}}{\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2\right)+{\displaystyle \frac{1}{\mathrm{p}+1}}\mathrm{K}({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})\hfill \\ & \ge {\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}\Vert ({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}}){\Vert }^2,\hfill \end{array}$$

we can determine that

$${\int }_0^{\mathrm{t}}{\Vert ({\partial }_{\mathsf{\tau }}{\mathrm{u}}_{\mathrm{m}},{\partial }_{\mathsf{\tau }}{\mathrm{v}}_{\mathrm{m}})\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }+{\displaystyle \frac{\mathrm{p}-1}{2(\mathrm{p}+1)}}{\Vert ({\mathrm{u}}_{\mathrm{m}},{\mathrm{v}}_{\mathrm{m}})\Vert }^2<\mathrm{d},\hspace{1em}0\le \mathrm{t}<\mathrm{\infty }.$$

The same logic applies to the rest.

Step 3. Proof when asymptotic behavior exists.

If $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}$ and $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$, based on Proposition 1 $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\in \mathrm{W}$, i.e., $\mathrm{E}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<\mathrm{d},\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)>0$ for $0\le \mathrm{t}<\mathrm{\infty }$ is obtained. If $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{d}$ and $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)\ge 0$, from the approximative solution $\left({\mathrm{u}}_{\mathrm{m}}\left(\mathrm{t}\right),{\mathrm{v}}_{\mathrm{m}}\left(\mathrm{t}\right)\right)\in \mathrm{W},$ we can deduce that $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\in \overline{\mathrm{W}}$, i.e., $\mathrm{E}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\le \mathrm{d},\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\ge 0$ for $0\le \mathrm{t}<\mathrm{\infty }$ is obtained. Hence, the definition of ${\mathrm{C}}^{\mathrm{*}}$ implies that

$$\mathrm{d}\ge \mathrm{E}(\mathrm{u},\mathrm{v})={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}\left(\Vert \nabla \mathrm{u}{\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2+\Vert \nabla \mathrm{v}{\Vert }_{{\mathrm{L}}^2\left(\mathsf{\Omega }\right)}^2\right)+{\displaystyle \frac{1}{\mathrm{p}+1}}\mathrm{K}(\mathrm{u},\mathrm{v})\ge {\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}\Vert (\mathrm{u},\mathrm{v}){\Vert }^2.$$

Therefore, from the definition of ${\mathrm{C}}_{\mathrm{*}}$ we can obtain

$$\begin{array}{c}\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}\left|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}\right|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\int }_{\mathsf{\Omega }}\left(|\mathrm{u}{|}^{\mathrm{p}+1}+|\mathrm{v}{|}^{\mathrm{p}+1}\right)\mathrm{d}\mathrm{x}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\left(\Vert \mathrm{u}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2+\Vert \mathrm{v}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2\right)}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\Vert (\mathrm{u},\mathrm{v}){\Vert }^{\mathrm{p}+1}\hfill \\ \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}{\left[{\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\mathrm{d}\right]}^{{\displaystyle \frac{\mathrm{p}-1}{2}}}\Vert (\mathrm{u},\mathrm{v}){\Vert }^2.\hfill \end{array}$$

We set $\mathsf{\sigma }:=2^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}{\left({\displaystyle \frac{\mathrm{p}+1}{\mathrm{p}-1}}\mathrm{d}\right)}^{{\displaystyle \frac{\mathrm{p}-1}{2}}}$ from Lemma 5, and we have $0<\mathsf{\sigma }<1$. Hence, for $0<\mathsf{\gamma }=1-\mathsf{\sigma }<1$, we obtain

$$\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\le \left(1-\mathsf{\gamma }\right)\Vert \left(\mathrm{u},\mathrm{v}\right){\Vert }^2,$$

i.e.,

$$\mathsf{\gamma }\Vert (\mathrm{u},\mathrm{v}){\Vert }^2\le \mathrm{K}(\mathrm{u},\mathrm{v})$$

(27)

Let $\mathrm{T}>0$ be a fixed time, since $(\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t}))\in \overline{\mathrm{W}}$ for $0\le \mathrm{t}<\mathrm{\infty }$ represents Equation (1)’s global WS:

$$\begin{array}{cc}{({\mathrm{u}}_{\mathrm{t}},\mathrm{u})}_2+{(\nabla {\mathrm{u}}_{\mathrm{t}},\nabla \mathrm{u})}_2+{(\nabla \mathrm{u},\nabla \mathrm{u})}_2& ={\left({\mathrm{f}}_1\right(\mathrm{u},\mathrm{v}),\mathrm{u})}_2,\\ {({\mathrm{v}}_{\mathrm{t}},\mathrm{v})}_2+{(\nabla {\mathrm{v}}_{\mathrm{t}},\nabla \mathrm{v})}_2+{(\nabla \mathrm{v},\nabla \mathrm{v})}_2& ={\left({\mathrm{f}}_2\right(\mathrm{u},\mathrm{v}),\mathrm{v})}_2,\end{array}$$

which means that

$${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2=-2\mathrm{K}(\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t}))\le 0,$$

i.e., $\mathrm{y}\left(\mathrm{t}\right)=\Vert \left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2$ is a non-increasing function for $0\le \mathrm{t}<\mathrm{\infty }$.

Hence, based on the Poincaré inequality and (27), we can deduce that

$$\begin{gathered} {\int }_{\mathrm{s}}^{\mathrm{T}}\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathrm{t}\le 2{\mathrm{d}}_{\mathsf{\Omega }}^2{\int }_{\mathrm{s}}^{\mathrm{T}}\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }^2\mathrm{d}\mathrm{t}\le {\displaystyle \frac{2{\mathrm{d}}_{\mathsf{\Omega }}^2}{\mathsf{\gamma }}}{\int }_{\mathrm{s}}^{\mathrm{T}}\mathrm{K}(\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t}))\mathrm{d}\mathrm{t} \\ \le {\displaystyle \frac{{\mathrm{d}}_{\mathsf{\Omega }}^2}{\mathsf{\gamma }}}\left(\Vert (\mathrm{u}(\mathrm{s}),\mathrm{v}(\mathrm{s})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2-\Vert (\mathrm{u}(\mathrm{T}),\mathrm{v}(\mathrm{T})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\right)\le {\displaystyle \frac{{\mathrm{d}}_{\mathsf{\Omega }}^2}{\mathsf{\gamma }}}\Vert (\mathrm{u}(\mathrm{s}),\mathrm{v}(\mathrm{s})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2 \end{gathered}$$

where $\mathrm{d}\left(\mathsf{\Omega }\right)$ represents the bounded domain’s diameter, $\mathsf{\Omega }$. Based on the arbitrary $\mathrm{T}>0$, it follows that

$${\int }_{\mathrm{s}}^{+\mathrm{\infty }}\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathrm{t}\le {\displaystyle \frac{{\mathrm{d}}_{\mathsf{\Omega }}^2}{\mathsf{\gamma }}}\Vert (\mathrm{u}(\mathrm{s}),\mathrm{v}(\mathrm{s})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2$$

for $0\le \mathrm{s}<\mathrm{\infty }$, which means (based on Lemma 11) that

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\le \Vert ({\mathrm{u}}_0,{\mathrm{v}}_0){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2{\mathrm{e}}^{1-{\displaystyle \frac{\mathrm{t}}{\mathrm{A}}}}$$

for $0\le \mathrm{t}<\mathrm{\infty }$ where $\mathrm{A}={\displaystyle \frac{1}{2\mathsf{\gamma }}}{\mathrm{d}}_{\mathsf{\Omega }}^2>0$. Hence,

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}\le {‖{\mathrm{u}}_0,{\mathrm{v}}_0‖}_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}{\mathrm{e}}^{{\displaystyle \frac{1}{2}}-\left(1-\mathsf{\sigma }\right){\mathrm{d}}_{\mathsf{\Omega }}^{-2}\mathrm{t}}, \mathrm{f}\mathrm{o}\mathrm{r} 0\le \mathrm{t}<\mathrm{\infty }.$$

□

Subsequently, the PW family case is accounted for by the global existence.

Corollary 1.

Let ${\mathrm{u}}_0,{\mathrm{v}}_0\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$. Assume that $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{d}$ and ${\mathrm{K}}_{{\mathrm{\delta }}_2}({\mathrm{u}}_0,{\mathrm{v}}_0)\ge 0$, where ${\mathrm{\delta }}_1<{\mathrm{\delta }}_2$ represents the equation’s two roots $\mathrm{d}\left(\mathrm{\delta }\right)=\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)$. The global WS of Equation (1) is identified, $\mathrm{u},\mathrm{v}\in$${\mathrm{L}}^{\infty }\left(0,\infty ;{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\right)$ with ${\mathrm{u}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}\in {\mathrm{L}}^2\left(0,\infty ;{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\right)$ and $\left(\mathrm{u}\left(\mathrm{t}\right),\mathrm{v}\left(\mathrm{t}\right)\right)\in {\mathrm{W}}_{\mathrm{\delta }}$ for ${\mathrm{\delta }}_1<\mathrm{\delta }<{\mathrm{\delta }}_2,\mathrm{t}\in [0,\infty )$.

Proof. 

Based on Theorem 1 and Proposition 1, it suffices to present $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$ using ${\mathrm{K}}_{{\mathsf{\delta }}_2}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$ to prove Corollary 1. If it is not true, $\overline{\mathsf{\delta }}\in \left[1,{\mathsf{\delta }}_2\right)$ exists such that ${\mathrm{K}}_{\overline{\mathsf{\delta }}}({\mathrm{u}}_0,{\mathrm{v}}_0)=0$. Thus, $({\mathrm{u}}_0,{\mathrm{v}}_0)\ne \left(0,0\right)$ implies that ${\mathrm{K}}_{{\mathsf{\delta }}_2}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$, and we obtain $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\ge \mathrm{d}\left(\overline{\mathsf{\delta }}\right)$. Yet, from Lemma 6, we have $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{d}\left({\mathsf{\delta }}_1\right)=\mathrm{d}\left({\mathsf{\delta }}_2\right)<\mathrm{d}\left(\mathsf{\delta }\right)$ for $\mathsf{\delta }\in \left({\mathsf{\delta }}_1,{\mathsf{\delta }}_2\right)$. This is a contradiction with ${\mathrm{K}}_{\overline{\mathsf{\delta }}}({\mathrm{u}}_0,{\mathrm{v}}_0)=0$; hence, the proof is completed. □

Instead of considering the outcome of the global existence that relies upon $\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)$, Equation (1)’s global existence was studied based on the initially available data ${\mathrm{u}}_0,{\mathrm{v}}_0$, relying on the ${\mathrm{H}}_0^1$ norm.

Corollary 2.

Let ${\mathrm{u}}_0,{\mathrm{v}}_0\in {\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)$. Presume that $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)\le \mathrm{d}$ and $\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0)\Vert <\mathrm{r}\left({\mathrm{\delta }}_2\right)$, where ${\mathrm{\delta }}_1<{\mathrm{\delta }}_2$ denote the equation’s two roots $\mathrm{d}\left(\mathrm{\delta }\right)=\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)$. Thus, the global WS of Equation (1) is identified, $\mathrm{u},\mathrm{v}\in {\mathrm{L}}^{\infty }\left(0,\infty ;{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right)\right),$ with ${\mathrm{u}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}\in {\mathrm{L}}^2(0,\infty ;{\mathrm{H}}_0^1\left(\mathrm{\Omega }\right))$ satisfying

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }^2<{\displaystyle \frac{\mathrm{E}\left({\mathrm{u}}_0,{\mathrm{v}}_0\right)}{\mathrm{a}\left({\mathsf{\delta }}_1\right)}},\mathrm{t}\in [0,\mathrm{\infty }).$$

(28)

Proof. 

From $\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0)\Vert <\mathrm{r}\left({\mathsf{\delta }}_2\right)$, we obtain ${\mathrm{K}}_{{\mathsf{\delta }}_2}({\mathrm{u}}_0,{\mathrm{v}}_0)>0$ using Lemma 2. Hence, problem (1) has a global $\mathrm{u},\mathrm{v}\in \mathrm{C}(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ with ${\mathrm{u}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}}\in {\mathrm{L}}^2(0,\mathrm{\infty };{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right))$ and $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\in {\mathrm{W}}_{\mathsf{\delta }}$ for ${\mathsf{\delta }}_1<\mathsf{\delta }<{\mathsf{\delta }}_2,\mathrm{t}\in [0,\mathrm{\infty })$ from Corollary 1. Eventually, (28) is followed by Lemma 8. □

3.2. The Solution’s Asymptotic Behavior with a Lower Bound on the Maximal Existence Time

Theorem 2 is proven. The PW’s subsequent property is important to proving it.

Lemma 12.

For any $\mathrm{ϵ}>0$, we have

$$\mathrm{d}\le {\mathrm{d}}_{\mathrm{ϵ}}+{\displaystyle \frac{2\mathrm{ϵ}}{\mathrm{p}+1}},$$

where ${\mathrm{d}}_{\mathrm{ϵ}}$ is defined as

$${\mathrm{d}}_{\mathsf{ϵ}}≔\mathrm{i}\mathrm{n}\mathrm{f}\left\{\mathrm{E}\left(\mathrm{u},\mathrm{v}\right)\mid \left(\mathrm{u},\mathrm{v}\right)\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right),\mathrm{K}\left(\mathrm{u},\mathrm{v}\right)=-\mathsf{ϵ}\right\}.$$

(29)

Proof. 

First, we show that, for any fixed $\mathsf{ϵ}>0, {\mathrm{d}}_{\mathsf{ϵ}}>-\mathrm{\infty }$.

Let $(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$ satisfy $\mathrm{K}(\mathrm{u},\mathrm{v})=-\mathsf{ϵ}<0$. Based on the definition of ${\mathrm{C}}_{\mathrm{*}}\left(9\right)$, we can deduce that

$$\begin{array}{cc}\hfill {\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}& \le \mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\int }_{\mathsf{\Omega }}\left(|\mathrm{u}{|}^{\mathrm{p}+1}+|\mathrm{v}{|}^{\mathrm{p}+1}\right)\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\left[{\left({\int }_{\mathsf{\Omega }}|\mathrm{u}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}\right)}^{{\displaystyle \frac{2}{\mathrm{p}+1}}}+{\left({\int }_{\mathsf{\Omega }}|\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}\right)}^{{\displaystyle \frac{2}{\mathrm{p}+1}}}\right]}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}{\left[{\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}\right]}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}.\hfill \end{array}$$

Thus,

$${\int }_{\mathsf{\Omega }}\left(\right|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2)\mathrm{d}\mathrm{x}\ge {\left[\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\right]}^{-{\displaystyle \frac{2}{\mathrm{p}-1}}}>0,$$

and

$$\begin{array}{cc}\hfill \mathrm{E}(\mathrm{u},\mathrm{v})& ={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\int }_{\mathsf{\Omega }}\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)\mathrm{d}\mathrm{x}+{\displaystyle \frac{1}{\mathrm{p}+1}}\mathrm{K}(\mathrm{u},\mathrm{v})\hfill \\ & \ge {\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\left[\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\right]}^{-{\displaystyle \frac{2}{\mathrm{p}-1}}}-{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{p}+1}}.\hfill \end{array}$$

Therefore,

$${\mathrm{d}}_{\mathsf{ϵ}}\ge {\displaystyle \frac{\mathrm{p}-1}{2(\mathrm{p}+1)}}{\left[\right(\mathrm{p}+1\left)\right(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\left){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\right]}^{-{\displaystyle \frac{2}{\mathrm{p}-1}}}-{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{p}+1}}>-\mathrm{\infty }.$$

Now, we choose a sequence $({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\backslash \left\{\left(\mathrm{0,0}\right)\right\}$ satisfying the following:

$$\mathrm{K}({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})=-\mathsf{ϵ},\mathrm{E}({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})\to {\mathrm{d}}_{\mathsf{ϵ}} \mathrm{a}\mathrm{s} \mathrm{j}\to \mathrm{\infty }.$$

Moreover, we also can suppose that $\mathrm{E}({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})$ is decreasing.

For each $({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})$, a ${\mathsf{\lambda }}_{\mathrm{j}}\in \mathbb{R}$ can be selected such that $\mathrm{K}({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}},{{\mathsf{\lambda }}_{\mathrm{j}}\mathrm{v}}_{\mathrm{j}})=0$. The value of ${\mathsf{\lambda }}_{\mathrm{j}}$ could be found specifically according to

$${\mathsf{\lambda }}_{\mathrm{j}}^{\mathrm{p}-1}={\displaystyle \frac{{\mathrm{M}}_{\mathrm{j}}}{{\mathrm{N}}_{\mathrm{j}}}},$$

where

$$\begin{gathered} {\mathrm{M}}_{\mathrm{j}}={\int }_{\mathsf{\Omega }}\left({\left|\nabla {\mathrm{u}}_{\mathrm{j}}\right|}^2+{\left|\nabla {\mathrm{v}}_{\mathrm{j}}\right|}^2\right)\mathrm{d}\mathrm{x} \\ {\mathrm{N}}_{\mathrm{j}}=\mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}{\left|{\mathrm{u}}_{\mathrm{j}}+{\mathrm{v}}_{\mathrm{j}}\right|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}{\left|{\mathrm{u}}_{\mathrm{j}}{\mathrm{v}}_{\mathrm{j}}\right|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}. \end{gathered}$$

Since $\mathrm{K}({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})=-\mathsf{ϵ}$ and $\mathrm{E}({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})\to {\mathrm{d}}_{\mathsf{ϵ}}$ as $\mathrm{j}\to \mathrm{\infty }$, we have

$${\mathrm{M}}_{\mathrm{j}}={\mathrm{N}}_{\mathrm{j}}-\mathsf{ϵ}$$

and

$$\mathrm{E}({\mathrm{u}}_{\mathrm{j}},{\mathrm{v}}_{\mathrm{j}})={\displaystyle \frac{1}{2}}{\mathrm{M}}_{\mathrm{j}}-{\displaystyle \frac{1}{\mathrm{p}+1}}{\mathrm{N}}_{\mathrm{j}}={\mathrm{d}}_{\mathsf{ϵ}}+{\mathsf{\eta }}_{\mathrm{j}}$$

where ${\mathsf{\eta }}_{\mathrm{j}}\to 0^{+}$, as $\mathrm{j}\to \mathrm{\infty }$.

Then, using ${\mathrm{d}}_{\mathsf{ϵ}}$ and ${\mathsf{\eta }}_{\mathrm{j}},{\mathrm{M}}_{\mathrm{j}}$ and ${\mathrm{N}}_{\mathrm{j}}$ can be expressed as

$${\mathrm{M}}_{\mathrm{j}}={\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\left({\mathrm{d}}_{\mathsf{ϵ}}+{\mathsf{\eta }}_{\mathrm{j}}+{\displaystyle \frac{2\mathsf{ϵ}}{\mathrm{p}+1}}\right),{\mathrm{N}}_{\mathrm{j}}={\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}\left({\mathrm{d}}_{\mathsf{ϵ}}+{\mathsf{\eta }}_{\mathrm{j}}+{\displaystyle \frac{1}{2}}\mathsf{ϵ}\right).$$

Notice that $\mathrm{K}({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}},{{\mathsf{\lambda }}_{\mathrm{j}}\mathrm{v}}_{\mathrm{j}})=0$, i.e., $({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}},{{\mathsf{\lambda }}_{\mathrm{j}}\mathrm{v}}_{\mathrm{j}})\in \mathcal{N}$. Hence, for all $\mathrm{j}\in \mathbb{N},\mathrm{E}({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}},{{\mathsf{\lambda }}_{\mathrm{j}}\mathrm{v}}_{\mathrm{j}})\ge \mathrm{d}$, a direct computation shows that

$$\begin{array}{cc}\hfill \mathrm{d}\le \mathrm{E}\left({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}},{\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{v}}_{\mathrm{j}}\right)& ={\displaystyle \frac{1}{2}}{\mathsf{\lambda }}_{\mathrm{j}}^2{\mathrm{M}}_{\mathrm{j}}-{\displaystyle \frac{1}{\mathrm{p}+1}}{\mathsf{\lambda }}_{\mathrm{j}}^{\mathrm{p}+1}{\mathrm{N}}_{\mathrm{j}}={\displaystyle \frac{1}{2}}{\left({\displaystyle \frac{{\mathrm{M}}_{\mathrm{j}}}{{\mathrm{N}}_{\mathrm{j}}}}\right)}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}{\mathrm{M}}_{\mathrm{j}}-{\displaystyle \frac{1}{\mathrm{p}+1}}{\left({\displaystyle \frac{{\mathrm{M}}_{\mathrm{j}}}{{\mathrm{N}}_{\mathrm{j}}}}\right)}^{{\displaystyle \frac{\mathrm{p}+1}{\mathrm{p}-1}}}{\mathrm{N}}_{\mathrm{j}}\hfill \\ & ={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\left({\displaystyle \frac{{\mathrm{M}}_{\mathrm{j}}}{{\mathrm{N}}_{\mathrm{j}}}}\right)}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}{\mathrm{M}}_{\mathrm{j}}={\left({\displaystyle \frac{{\mathrm{d}}_{\mathsf{ϵ}}+{\mathsf{\eta }}_{\mathrm{j}}+\frac{\mathsf{ϵ}}{\mathrm{p}+1}}{{\mathrm{d}}_{\mathsf{ϵ}}+{\mathsf{\eta }}_{\mathrm{j}}+\frac{1}{2}\mathsf{ϵ}}}\right)}^{{\displaystyle \frac{2}{\mathrm{p}-1}}}\left({\mathrm{d}}_{\mathsf{ϵ}}+{\mathsf{\eta }}_{\mathrm{j}}+{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{p}+1}}\right)\hfill \\ & \le {\mathrm{d}}_{\mathsf{ϵ}}+{\mathsf{\eta }}_{\mathrm{j}}+{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{p}+1}}.\hfill \end{array}$$

From ${\mathsf{\eta }}_{\mathrm{j}}\to 0^{+}$ and $\mathrm{j}\to \mathrm{\infty },$ we have $0<{\mathsf{\eta }}_{\mathrm{j}}<{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{p}+1}}$ for sufficiently large values of $\mathrm{j}$.

Therefore,

$$\mathrm{d}\le {\mathrm{d}}_{\mathsf{ϵ}}+{\displaystyle \frac{2\mathsf{ϵ}}{\mathrm{p}+1}}.$$

□

Theorem 2 will now be proven.

Proof of Theorem 2.

Two steps are used.

Step 1: Proof when the case initially has low energy.

For $({\mathrm{u}}_0,{\mathrm{v}}_0)\in \mathrm{Z}$, we choose $\mathsf{ϵ}>0,$ such that

$$\mathsf{ϵ}<\min \{{\displaystyle \frac{1}{2}}(\mathrm{d}-\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)),-\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)\}.$$

$\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)<-\mathsf{ϵ}$ due to the selection of $\mathsf{ϵ}$.

Assume that $\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<-\mathsf{ϵ}$ for all $\mathrm{t}\in (0,\mathrm{T})$. $\mathrm{T}>0$ shows the highest existence duration. Alternatively, if $\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)$ is time continuous, there exists ${\mathrm{t}}^{\mathrm{*}}\in \left(0,\mathrm{T}\right),$ which satisfies $\mathrm{K}\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)=-\mathsf{ϵ}$. Based on Lemma 12, we know that

$$\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)<\mathrm{d}-\mathsf{ϵ}<\mathrm{d}-{\displaystyle \frac{2\mathsf{ϵ}}{\mathrm{p}+1}}\le {\mathrm{d}}_{\mathsf{ϵ}}.$$

Due to energy conservation (7), $\mathrm{E}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<{\mathrm{d}}_{\mathsf{ϵ}}$ for all $\mathrm{t}\in (0,\mathrm{T})$ is obtained.

In addition, $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}^{\mathrm{*}}),\mathrm{v}({\mathrm{t}}^{\mathrm{*}}\left)\right)<{\mathrm{d}}_{\mathsf{ϵ}}$, which contradicts the definition of ${\mathrm{d}}_{\mathsf{ϵ}}\left(29\right)$. For any $(\mathrm{u},\mathrm{v})\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\times {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\backslash \left\{\left(\mathrm{0,0}\right)\right\}$, let

$$\mathrm{L}\left(\mathrm{t}\right)={\displaystyle \frac{1}{2}}\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2.$$

On this basis, we have ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\mathrm{L}(\mathrm{t})=-\mathrm{K}(\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t}))>\mathsf{ϵ}$ for $\mathrm{t}\in (0,\mathrm{T})$. Integrating from 0 to $\mathrm{t}$, we can determine that

$$\mathrm{L}\left(\mathrm{t}\right)\ge {\int }_0^{\mathrm{t}}\mathsf{ϵ}\mathrm{d}\mathrm{s}+\mathrm{L}\left(0\right)=\mathsf{ϵ}\mathrm{t}+\mathrm{L}\left(0\right),\hspace{1em}\mathrm{f}\mathrm{o}\mathrm{r}\hspace{1em}\mathrm{t}\in (0,\mathrm{T}).$$

(30)

Hence, for $\mathrm{t}\in (0,\mathrm{T})$, we have

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}\ge {\left(\mathsf{ϵ}\mathrm{t}+\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

Along the flow generated by Equation (1), based on the definition of ${\mathrm{C}}_{\mathrm{*}}\left(9\right)$, we can deduce that

$$\begin{array}{cc}\hfill {\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\mathrm{L}\left(\mathrm{t}\right)& ={\int }_{\mathsf{\Omega }}\left[\mathrm{u}\left({\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}}\right)+\mathrm{v}\left({\mathrm{v}}_{\mathrm{t}}-\mathsf{\Delta }{\mathrm{v}}_{\mathrm{t}}\right)\right]\mathrm{d}\mathrm{x}\hfill \\ & =-{\int }_{\mathsf{\Omega }}\left[\left(|\nabla \mathrm{u}{|}^2+|\nabla \mathrm{v}{|}^2\right)-\left(\mathrm{p}+1\right)\mathsf{\alpha }\left|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}-\left(\mathrm{p}+1\right)2\mathsf{\beta }\right|\mathrm{u}\mathrm{v}{|}^{\left(\mathrm{p}+1|2\right)}\right]\mathrm{d}\mathrm{x}\hfill \\ & =-\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\hfill \\ & \le \mathsf{\alpha }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}+\mathrm{v}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}+2\mathsf{\beta }\left(\mathrm{p}+1\right){\int }_{\mathsf{\Omega }}|\mathrm{u}\mathrm{v}{|}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\int }_{\mathsf{\Omega }}\left(|\mathrm{u}{|}^{\mathrm{p}+1}+|\mathrm{v}{|}^{\mathrm{p}+1}\right)\mathrm{d}\mathrm{x}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\left(\Vert \mathrm{u}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2+\Vert \mathrm{v}{\Vert }_{{\mathrm{L}}^{\mathrm{p}+1}\left(\mathsf{\Omega }\right)}^2\right)}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\hfill \\ & \le \left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\Vert (\mathrm{u},\mathrm{v}){\Vert }^{\mathrm{p}+1}\hfill \\ & \le 2^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\mathrm{L}{\left(\mathrm{t}\right)}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}.\hfill \end{array}$$

Since $({\mathrm{u}}_0,{\mathrm{v}}_0)\in \mathrm{Z}$, from Proposition 2, we know that $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)\in \mathrm{Z}$, i.e., $\mathrm{E}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<\mathrm{d}$ and $\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<0$ for $\mathrm{t}\in (0,\mathrm{T})$. It follows that ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\mathrm{L}(\mathrm{t})=-\mathrm{K}(\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t}))>0$, i.e., $\mathrm{L}\left(\mathrm{t}\right)$ increases as Equation (1) generates the flow and $\mathrm{L}\left(\mathrm{t}\right)>\mathrm{L}\left(0\right)\ge 0$ for $\mathrm{t}\in (0,\mathrm{T})$. Therefore, we can deduce that

$${\displaystyle \frac{\frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}\mathrm{L}\left(\mathrm{t}\right)}{\mathrm{L}{\left(\mathrm{t}\right)}^{\frac{\mathrm{p}+1}{2}}}}\le 2^{{\displaystyle \frac{\mathrm{p}+1}{2}}}\left(\mathrm{p}+1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}.$$

(31)

By integrating the inequality (31) from 0 to $\mathrm{t},$ it follows that

$$\mathrm{L}{\left(0\right)}^{-{\displaystyle \frac{\mathrm{p}-1}{2}}}-\mathrm{L}{\left(\mathrm{t}\right)}^{-{\displaystyle \frac{\mathrm{p}-1}{2}}}\le 2^{{\displaystyle \frac{\mathrm{p}-1}{2}}}\left({\mathrm{p}}^2-1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}\mathrm{t}.$$

Hence, if $0<\mathrm{t}<{\displaystyle \frac{1}{\mathsf{\mu }}}{\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0)\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^{-(\mathrm{p}-1)}$, we can deduce that

$$\Vert (\mathrm{u}(\mathrm{t}),\mathrm{v}(\mathrm{t})){\Vert }_{{\mathrm{H}}_0^1(\mathsf{\Omega })}\le {({\Vert ({\mathrm{u}}_0,{\mathrm{v}}_0)\Vert }_{{\mathrm{H}}_0^1(\mathsf{\Omega })}^{-(\mathrm{p}-1)}-\mathsf{\mu }\mathrm{t})}^{-{\displaystyle \frac{1}{\mathrm{p}-1}}}<+\mathrm{\infty }$$

where $\mathsf{\mu }=2^{\mathrm{p}-1}\left({\mathrm{p}}^2-1\right)\left(2^{\mathrm{p}}\mathsf{\alpha }+\mathsf{\beta }\right){\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}$.

Step 2: Proof for an initially overcritical energy case.

Let $\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)$ denote Equation (1)’s WS for $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{d}>0,\mathrm{K}({\mathrm{u}}_0,{\mathrm{v}}_0)<0$.

Based on the time continuity of $\mathrm{E}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)$ and $\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)$, a sufficiently small ${\mathrm{t}}_0>0$ exists such that $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)>0$ and $\mathrm{K}\left(\mathrm{u}\right(\mathrm{t}),\mathrm{v}(\mathrm{t}\left)\right)<0$ for $0\le \mathrm{t}\le {\mathrm{t}}_0$.

Thus, we can deduce that

$${\int }_{\mathsf{\Omega }}\left[\mathrm{u}\right({\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}})+\mathrm{v}({\mathrm{v}}_{\mathrm{t}}-\mathrm{v}\mathsf{\Delta }{\mathrm{v}}_{\mathrm{t}}\left)\right]\mathrm{d}\mathrm{x}=-\mathrm{K}(\mathrm{u},\mathrm{v})>0$$

and ${\Vert ({\mathrm{u}}_{\mathrm{t}},{\mathrm{v}}_{\mathrm{t}})\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2>0$ for $0\le \mathrm{t}\le {\mathrm{t}}_0$. Therefore, ${\int }_0^{\mathrm{t}}\Vert ({\mathrm{u}}_{\mathsf{\tau }},{\mathrm{v}}_{\mathsf{\tau }}){\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }$ strictly increases upon $0\le \mathrm{t}\le {\mathrm{t}}_0$, and ${\mathrm{t}}_0>0$ is chosen such that

$$0<{\mathrm{d}}_0=\mathrm{d}-{\int }_0^{{\mathrm{t}}_0}{\Vert ({\mathrm{u}}_{\mathsf{\tau }},{\mathrm{v}}_{\mathsf{\tau }})\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }<\mathrm{d}.$$

The equation $\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)=\mathrm{d}$ and the energy conservation (7), as well as the previous inequality, suggest that

$$\mathrm{E}(\mathrm{u}({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0))=\mathrm{E}({\mathrm{u}}_0,{\mathrm{v}}_0)-{\int }_0^{{\mathrm{t}}_0}{\Vert ({\mathrm{u}}_{\mathsf{\tau }},{\mathrm{v}}_{\mathsf{\tau }})\Vert }_{{\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)}^2\mathrm{d}\mathsf{\tau }<\mathrm{d}.$$

When $\mathrm{t}={\mathrm{t}}_0$ is taken as the initial time, $\mathrm{E}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)<\mathrm{d}$ and $\mathrm{K}\left(\mathrm{u}\right({\mathrm{t}}_0),\mathrm{v}({\mathrm{t}}_0\left)\right)<0$.

Hence, we can determine from step 1 that the result follows when we choose $\mathrm{t}={\mathrm{t}}_0$ instead of $\mathrm{t}=0$ as the conditions’ initial times for $\mathrm{t}$ and $\mathsf{ϵ}$. □

4. Conclusions

This article focuses on analyzing the initial boundary value problem involving a coupled system of semi-linear pseudo-parabolic equations on a bounded smooth domain. By utilizing the potential well method, we explore the invariance of certain sets under the condition of the problem flow and the vacuum-isolating behavior of the solutions. Additionally, we demonstrate the global existence of solutions with exponential decay, as well as their asymptotic behavior, while establishing a lower bound on the maximal existence time for both low-initial-energy and critical-energy scenarios. A crucial aspect of our findings is the identification of a sharp condition that relates to these phenomena. Furthermore, we prove that global existence also applies to the case of a potential well family.

The problem considered here encompasses a variety of physically significant processes, such as the unidirectional propagation of nonlinear dispersive long waves [1,2], the seepage of homogeneous fluids through fissured rock [3], and the disparity between conductive and thermodynamic temperatures [4].

The inclusion of pseudo-parabolic terms is crucial to determining the lifespan, blasting set, and blasting rate of these equations. A thorough understanding of their specific impact is vital for analyzing the asymptotic behavior of solutions, making it a matter of considerable theoretical significance.

Our findings make a valuable contribution to the comprehension of coupled pseudo-parabolic systems and their long-term dynamics, an area of research that is both active and demanding in the fields of mathematical analysis and applied mathematics. Our innovative approach and the results put forth have the ability to spark future explorations in similar fields.