The Asymptotic Behavior and Blow-Up Rate of a Solution with a Lower Bound on the Highest Existence Duration for Semi-Linear Pseudo-Parabolic Equations

Abstract

This note addresses the initial-boundary value problem for a class of semi-linear pseudo-parabolic equations defined on a smooth bounded domain, with an emphasis on determining the asymptotic behavior and blow-up rate of the solution. Our analysis considers both low-initial energy and critical-initial energy cases, with a specific focus on establishing a lower bound on the maximal existence time of the solutions to this problem.

1. Introduction

The asymptotic behavior of the solution and its blow-up rate are investigated with a lower bound on the maximal existence duration to the following initial-boundary value problem (IBVP) for nonlinear pseudo-parabolic equations:

$$\left\{\begin{array}{cc}{\mathrm{u}}_{\mathrm{t}}-\mathsf{\Delta }\mathrm{u}-\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}}={\mathrm{u}}^{\mathrm{p}},& \mathrm{i}\mathrm{n}\mathsf{\Omega }\times \left(0,\mathrm{T}\right),\\ \mathrm{u}=0,& \mathrm{o}\mathrm{n}\mathsf{\Gamma }\times \left(0,\mathrm{T}\right),\\ \mathrm{u}\left(\mathrm{x},0\right)={\mathrm{u}}_0,& \mathrm{x}\in \mathsf{\Omega },\end{array}\right.$$

(1)

where $0<\mathrm{T}\le +\mathrm{\infty }$ is the maximal existence time, ${\mathrm{u}}_0\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$, $\mathsf{\Omega }$ denotes a bounded domain in ${\mathbb{R}}^{\mathrm{n}}$ with a smooth boundary $\partial \mathsf{\Omega }=\mathsf{\Gamma }$, and $\mathrm{p}$ satisfies the following:

$$(\mathrm{H})1<\mathrm{p}<\mathrm{\infty }\mathrm{i}\mathrm{f}\mathrm{n}=1,2;1<\mathrm{p}\le {\displaystyle \frac{\mathrm{n}+2}{\mathrm{n}-2}}\mathrm{i}\mathrm{f}\mathrm{n}\ge 3.$$

Equation (1) characterizes various physically significant procedures, such as unidirectionally propagated waves characterized as nonlinear, dispersive, and long [1,2] (where $u$ is typically the amplitude or velocity), and population aggregation [3] (where $u$ represents the population density). Equation (1) can be utilized to analyze the non-stationarity behavior of semiconductors when presented by sources [4,5].

This equation is classified as a Sobolev-type differential equation, distinguished by its incorporation of mixed time and space derivatives among the highest-order terms. Specifically, when the highest-order term is a single time derivative, the equation is referred to as a pseudo-parabolic equation. This concept can be traced back to the foundational work of Showalter in the 1970s [6]. Since then, there has been a notable increase in research on various models of pseudo-parabolic equations. A.I. Kozhanov, a prominent Russian expert in this area, has made significant contributions, as documented in [7] and other publications. Additionally, recently, a review of studies related to pseudo-parabolic equations was carried out by Cao and Yin [8].

The available outcomes regarding the initially conditioned boundary value problem are recalled concerning the semi-linear pseudo-parabolic equation

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

(2)

Cao et al. [9] investigated the Cauchy problem for an n-dimensional Equation (2) with the nonlinearity $\mathrm{f}\left(\mathrm{u}\right)={\mathrm{u}}^{\mathrm{p}}$. They employed integral representations alongside the contraction-mapping principle to demonstrate both the existence and uniqueness of solutions for all $\mathrm{p}>0$ while also establishing several related comparison principles. Moreover, they found that if $0<\mathrm{p}\le 1$, the solution to (2) exists globally. In contrast, for $1<\mathrm{p}<1+{\displaystyle \frac{2}{\mathrm{n}}}$, the solution experiences blow-up in a finite time for any positive nontrivial initial data. Furthermore, if $\mathrm{p}\ge 1+{\displaystyle \frac{2}{\mathrm{n}}}$, the solution also blows up in finite time, but this occurs specifically for sufficiently large positive initial data.

The potential well method was employed by R.Z. Xu et al. to examine Equation (2) [10]. The global existence of solutions with exponential decay and finite-time blow-up was proven by R.Z. Xu et al. [10] for a case wherein the initial energy was $\mathrm{E}\left({\mathrm{u}}_0\right)\le \mathrm{d}$ regarding initially conditioned boundary value problems represented by Equation (2) and $\mathrm{f}\left(\mathrm{u}\right)={\mathrm{u}}^{\mathrm{p}}$. More precisely, when $1<\mathrm{p}<\mathrm{\infty }$, the outcomes are attained if $\mathrm{n}=1,2$, or $1<\mathrm{p}\le {\displaystyle \frac{\mathrm{n}+2}{\mathrm{n}-2}}$ if $\mathrm{n}\ge 3$. Moreover, a finite-time blow-up with high initial energy $\mathrm{E}\left({\mathrm{u}}_0\right)>\mathrm{d}$ can be achieved via the comparison principle.

Based on this work, many results have been used to discuss the asymptotic behavior of pseudo-parabolic equations with source terms [11,12,13], variable index pseudo-parabolic equations [14], and degenerate singular pseudo-parabolic equations [15] using the potential well method.

Even though the finite-time blasting and global existence results of the solutions of semi-linear pseudo-parabolic equations have been discussed in detail previously, there are still many problems that remain unsolved, such as the exact influence that pseudo-parabolic terms have on the life span, the blasting set, and the blasting rate of pseudo-parabolic equations, which are of great theoretical value in studying the asymptotic behavior of solutions.

The main aim of this article is to contribute to solving the problems mentioned above. This study represents the first attempt to tackle Equation (1) in order to study problems with the term $\mathsf{\Delta }{\mathrm{u}}_{\mathrm{t}}$. With the exception of some preliminary results, we are able to determine the asymptotic behavior and blow-up rate of the solution. This analysis applies to both the low-initial energy case and the critical-initial energy case, with a focus on establishing a lower bound on the maximal existence time.

The potential well method used in this paper has wide applicability in asymptotic analysis and other areas of applied mathematics and beyond. For example, it can be well applied for extending the results from [16], where a highly conductive orebody buried in earth is investigated via a perfectly conducting triaxial model ellipsoid.

First, we present a description of the weak solution:

Definition 1

(Weak solution (WS)). Let $u=u\left(x,t\right)$, denoting Equation (1)’s WS for $\Omega \times [0,T)$ when $0<T\le +\infty$ denotes the highest existence duration if $u\in L^{\infty }(0,T;H_0^1\left(\Omega \right))$ with $u_t\in L^2(0,T;L^2\left(\Omega \right))$ and Equation (1) is satisfied distributionally, i.e.,

$${\left({\mathrm{u}}_{\mathrm{t}},\mathrm{w}\right)}_2+{\left(\nabla {\mathrm{u}}_{\mathrm{t}},\nabla \mathrm{w}\right)}_2+{\left(\nabla \mathrm{u},\nabla \mathrm{w}\right)}_2={\left({\mathrm{u}}^{\mathrm{p}},\mathrm{w}\right)}_2,$$

for any $w\in H_0^1\left(\Omega \right),t\in (0,T)$ with $u(x,0)=u_0$ for $x\in \Omega$, and ${(·,·)}_2$ shows the inner product (IP) ${(·,·)}_{L^2\left(\Omega \right)}$.

Problem (1) possesses a variational structure. Namely, for $u\in H_0^1\left(\Omega \right)$, the energy functionals of Problem (1) are given by the following:

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

(3)

Remark 1.

The energy conservation is satisfied by the definition of the weak solution presented above, i.e.,

$${\int }_0^t{\Vert {\partial }_{\tau }u\Vert }_{H_0^1\left(\Omega \right)}^{2}d\tau +E\left(u\right)=E\left(u_0\right),fort\ge 0.$$

(4)

For $u\in H_0^1\left(\Omega \right)$, the Nehari functional is defined by

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

(5)

From the embedding $H_0^1\left(\Omega \right)\hookrightarrow L^{p+1}\left(\Omega \right)$ and the Poincaré inequality, we can define the following:

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

(6)

The potential well depth $d$ is delineated by

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

(7)

where

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

represents the Nehari manifold.

Considering

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

the potential well method is proposed, and the potential well methods outside are delineated by means of the following set:

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

The subsequent sections of the article are organized as follows: Section 2 presents a preliminary result essential for proving Theorem 1 in Section 3, which is the primary focus of this article. Section 4 concludes the paper.

2. Preliminary Result

This section establishes a preliminary result necessary for demonstrating the statement of Theorem 1 in Section 3.

Lemma 1.

Consider $u_0\in H_0^1\left(\Omega \right)$, where $\Omega$ is a bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\partial \Omega =\Gamma$, and $p$ satisfies

$$(H)1<p<\infty ifn=1,2;1<p\le {\displaystyle \frac{n+2}{n-2}}ifn\ge 3.$$

For any $ϵ>0$, we have

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

where $d_{ϵ}$ is defined by

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

(8)

Proof of Lemma 1.

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

Let $\mathrm{u}\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)$ satisfy $\mathrm{K}(\mathrm{u})=-\mathsf{ϵ}<0$. From the definition of ${\mathrm{C}}_{\mathrm{*}}\left(6\right)$, we obtain

$${\int }_{\mathsf{\Omega }}|\nabla \mathrm{u}{|}^2\mathrm{d}\mathrm{x}\le {\int }_{\mathsf{\Omega }}|\mathrm{u}{|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}\le {\mathrm{C}}_{\mathrm{*}}^{\mathrm{p}+1}{\int }_{\mathsf{\Omega }}|\nabla \mathrm{u}{|}^2\mathrm{d}{\mathrm{x}}^{{\displaystyle \frac{\mathrm{p}+1}{2}}}.$$

Hence,

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

followed by

$$\mathrm{E}(\mathrm{u})={\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\int }_{\mathsf{\Omega }}|\nabla \mathrm{u}{|}^2\mathrm{d}\mathrm{x}+{\displaystyle \frac{1}{\mathrm{p}+1}}\mathrm{K}(\mathrm{u})\ge {\displaystyle \frac{\mathrm{p}-1}{2\left(\mathrm{p}+1\right)}}{\mathrm{C}}_{\mathrm{*}}^{-{\displaystyle \frac{2\left(\mathrm{p}+1\right)}{\mathrm{p}-1}}}-{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{p}+1}}.$$

Thus,

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

where ${\mathrm{u}}_{\mathrm{j}}\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}$, a sequence of elements, is chosen in order to satisfy

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

Decreases in $\mathrm{E}\left({\mathrm{u}}_{\mathrm{j}}\right)$ are supposed. For every ${\mathrm{u}}_{\mathrm{j}},{\mathsf{\lambda }}_{\mathrm{j}}\in \mathbb{R}$, is chosen so that $\mathrm{K}\left({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}}\right)=0$. In fact, ${\mathsf{\lambda }}_{\mathrm{j}}$ can be explicitly found using the following:

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

where

$${\mathrm{M}}_{\mathrm{j}}={\int }_{\mathsf{\Omega }}{\left|\nabla {\mathrm{u}}_{\mathrm{j}}\right|}^2\mathrm{d}\mathrm{x},{\mathrm{N}}_{\mathrm{j}}={\int }_{\mathsf{\Omega }}{\left|{\mathrm{u}}_{\mathrm{j}}\right|}^{\mathrm{p}+1}\mathrm{d}\mathrm{x}.$$

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

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

and

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

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 by

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

One can observe that $\mathrm{K}\left({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}}\right)=0$, i.e., ${\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}}\in \mathcal{N}$. Hence, for all $\mathrm{j}\in \mathbb{N},\mathrm{E}\left({\mathsf{\lambda }}_{\mathrm{j}}{\mathrm{u}}_{\mathrm{j}}\right)\ge \mathrm{d}$, a direct computation gives the following:

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

Based on ${\mathsf{\eta }}_{\mathrm{j}}\to 0^{+}$ as $\mathrm{j}\to \mathrm{\infty },0<{\mathsf{\eta }}_{\mathrm{j}}<{\displaystyle \frac{\mathsf{ϵ}}{\mathrm{p}+1}}$ holds when $\mathrm{j}$ becomes sufficiently large. Thus,

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

 □

3. Main Result

The main focus of this article is on the following outcomes, which characterize the asymptotic behavior of the solution and its blow-up rate in Problem (1) for low- and critical-initial energy cases while also providing a lower bound on the maximal existence duration. R.Z. Xu et al. [10] obtained finite-time blow-up results for the solutions of Problem (1) with the conditions listed in the following theorem.

Theorem 1.

Consider $u_0\in H_0^1\left(\Omega \right)$, where $\Omega$ denotes a bounded domain in ${\mathbb{R}}^n$ with a smooth boundary $\partial \Omega =\Gamma$, and $p$ satisfies

$$(H)1<p<\infty ifn=1,2;1<p\le {\displaystyle \frac{n+2}{n-2}}ifn\ge 3.$$

Assume that $E(u_0)<d$ and $K(u_0)<0$, and the maximal existence time $T$ is bounded from below by

$$\mathrm{T}\ge {\displaystyle \frac{1}{\mathsf{\mu }}}{\Vert {\mathrm{u}}_0\Vert }_{{\mathrm{H}}_0^1(\mathsf{\Omega })}^{-(\mathrm{p}-1)}$$

where $\mu =2^{p-1}\left(p-1\right)C_{*}^{p+1}$ . Then, the weak solution of problem (1) satisfies the following:

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

for $0<t<{\displaystyle \frac{1}{\mu }}{\Vert u_0\Vert }_{H_0^1(\Omega )}^{-(p-1)}$.

Moreover, for all $0<t<T$, the weak solution of Problem (1) satisfies

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

where $0<ϵ<\mathit{min}\{{\displaystyle \frac{1}{2}}(d-E(u_0)),-K(u_0)\}$.

If $E(u_0)=d$ and $K(u_0)<0$, the result follows by choosing $t=t_0$ for a sufficiently small $t_0>0$ instead of the initial time $t=0$ for both $t$ and $ϵ$.

Proof of Theorem 1.

We split the proof into two main steps.

Step 1: The low-initial energy case.

For each ${\mathrm{u}}_0\in \mathrm{Z},\mathsf{ϵ}>0$,

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

$\mathrm{K}({\mathrm{u}}_0)<-\mathsf{ϵ}$ to satisfy the above statement.

We claim that $\mathrm{K}(\mathrm{u}(\mathrm{t}))<-\mathsf{ϵ}$ for every $\mathrm{t}\in \left(0,\mathrm{T}\right)$, where $\mathrm{T}>0$ is the longest existence time. Should this not hold, there should exist a ${\mathrm{t}}^{\mathrm{*}}\in \left(0,\mathrm{T}\right)$ satisfying $\mathrm{K}\left(\mathrm{u}\left({\mathrm{t}}^{\mathrm{*}}\right)\right)=-\mathsf{ϵ}$ when $\mathrm{K}\left(\mathrm{u}\left(\mathrm{t}\right)\right)$ is time-continuous. Based on Lemma 1, we know that

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

Then, the energy conversion Equation (4) leads to $\mathrm{E}(\mathrm{u}(\mathrm{t}))<{\mathrm{d}}_{\mathsf{ϵ}}$ for all $\mathrm{t}\in \left(0,\mathrm{T}\right)$. Hence, $\mathrm{E}(\mathrm{u}({\mathrm{t}}^{\mathrm{*}}))<{\mathrm{d}}_{\mathsf{ϵ}}$, which contracts the definition of ${\mathrm{d}}_{\mathsf{ϵ}}\left(8\right)$.

For any $\mathrm{u}\in {\mathrm{H}}_0^1\left(\mathsf{\Omega }\right)\backslash \left\{0\right\}$, let

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

Problem (1) suggests that

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

Based on the above, we have ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\mathrm{L}(\mathrm{t})=-\mathrm{K}(\mathrm{u}(\mathrm{t}))>\mathsf{ϵ}$ for $\mathrm{t}\in \left(0,\mathrm{T}\right)$. Integrating from 0 to $\mathrm{t}$, we obtain

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

(9)

Hence, if $0<\mathrm{t}<\mathrm{T}$, one obtains

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

Since ${\mathrm{u}}_0\in \mathrm{Z}$, based on the invariant sets of solutions to (1), we know that $\mathrm{u}\left(\mathrm{t}\right)\in \mathrm{Z}$, i.e., $\mathrm{E}(\mathrm{u}(\mathrm{t}))<\mathrm{d}$, and $\mathrm{K}(\mathrm{u}(\mathrm{t}))<0$ for $\mathrm{t}\in \left(0,\mathrm{T}\right)$, as determined by using the solution invariant sets of (1). Then, ${\mathrm{t}}_0\in \left(0,\mathrm{T}\right)$ exists such that $\mathrm{u}\left({\mathrm{t}}_0\right)\in \partial \mathrm{Z}$, which implies that $\mathrm{E}\left(\mathrm{u}\left({\mathrm{t}}_0\right)\right)=\mathrm{d}$, or $\mathrm{K}\left(\mathrm{u}\left({\mathrm{t}}_0\right)\right)=0$ if this cannot happen. Then, (4) leads to $\mathrm{E}(\mathrm{u}(\mathrm{t}))<\mathrm{E}({\mathrm{u}}_0)<\mathrm{d}$. Thus, we have $\mathrm{K}\left(\mathrm{u}\left({\mathrm{t}}_0\right)\right)=0$, so $\mathrm{d}={\mathrm{i}\mathrm{n}\mathrm{f}}_{\mathrm{u}\in \mathcal{N}}\mathrm{E}\left(\mathrm{u}\right)$ contradicts $\mathrm{E}(\mathrm{u}({\mathrm{t}}_0))>\mathrm{d}$.

From the definition of ${\mathrm{C}}_{\mathrm{*}}$(6), we can obtain

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

It follows that ${\displaystyle \frac{\mathrm{d}}{\mathrm{d}\mathrm{t}}}\mathrm{L}(\mathrm{t})=-\mathrm{K}(\mathrm{u}(\mathrm{t}))>0$, i.e., $\mathrm{L}\left(\mathrm{t}\right)$ increases along the flow produced by Problem (1), and $\mathrm{L}(\mathrm{t})>\mathrm{L}(0)\ge 0$ for $\mathrm{t}\in \left(0,\mathrm{T}\right)$.

Therefore, we obtain

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

and integrating the above inequality from 0 to $\mathrm{t}$ yields

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

Hence, if $0<\mathrm{t}<{\displaystyle \frac{1}{\mathsf{\mu }}}{\Vert {\mathrm{u}}_0\Vert }_{{\mathrm{H}}_0^1(\mathsf{\Omega })}^{-(\mathrm{p}-1)}$, one obtains

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

and note that the maximal existence time $\mathrm{T}$ is bounded from below by

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

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

Step 2: The critical-initial energy case.

Let $\mathrm{u}\left(\mathrm{t}\right)$ denote the weak solution of (1), where $\mathrm{E}({\mathrm{u}}_0)=\mathrm{d}>0,\mathrm{K}({\mathrm{u}}_0)<0$.

Since $\mathrm{E}\left(\mathrm{u}\left(\mathrm{t}\right)\right)$ and $\mathrm{K}\left(\mathrm{u}\left(\mathrm{t}\right)\right)$ are time-continuous, a sufficiently small ${\mathrm{t}}_0>0$ exists such that $\mathrm{E}(\mathrm{u}({\mathrm{t}}_0))>0$ and $\mathrm{K}(\mathrm{u}(\mathrm{t}))<0$ for $0\le \mathrm{t}\le {\mathrm{t}}_0$.

Thus, we can deduce that

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

and ${\Vert {\mathrm{u}}_{\mathrm{t}}\Vert }_{{\mathrm{H}}_0^1(\mathsf{\Omega })}^2>0$ for $0\le \mathrm{t}\le {\mathrm{t}}_0$.

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

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

(10)

Since $\mathrm{E}\left({\mathrm{u}}_0\right)=\mathrm{d}$, (4) and the inequality above lead to

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

When $\mathrm{t}={\mathrm{t}}_0$, it is observed that $\mathrm{E}(\mathrm{u}({\mathrm{t}}_0))<\mathrm{d}$ and $\mathrm{K}(\mathrm{u}({\mathrm{t}}_0))<0$.

Hence, from Step 1, we can infer that the assertion follows by initially choosing $\mathrm{t}={\mathrm{t}}_0$ instead of $\mathrm{t}=0$ when $\mathrm{t}$ and $\mathsf{ϵ}$ are conditioned. □

4. Conclusions

This article focuses on the initial-boundary value problem for a class of semi-linear pseudo-parabolic equations on a smooth bounded domain. We were able to determine the asymptotic behavior and blow-up rate of the solution to this problem. Our analysis encompasses both the low-initial energy case and the critical-initial energy case, with a particular focus on establishing a lower bound on the maximal existence time.

The considered problem captures various physically meaningful processes, such as the dynamics of nonlinear, dispersive, and long waves propagating unidirectionally [1,2] (with $\mathrm{u}$ typically denoting amplitude or velocity), as well as population aggregation [3] (with $\mathrm{u}$ representing population density). Utilizing Equation (1), one can study the non-stationarity behavior of semiconductors in the presence of sources [4,5].

The presence of pseudo-parabolic terms is pivotal in determining the life span, blasting set, and blasting rate of pseudo-parabolic equations. Understanding their precise impact is essential for analyzing the asymptotic behavior of solutions, rendering it a topic of significant theoretical importance.

We are confident that our findings provide a relevant contribution to the understanding and study of pseudo-parabolic equations and their long-term dynamics, which remains an active and challenging area of research in mathematical analysis and applied mathematics. Our innovative approach and results have the potential to spark further exploration in related fields.

Studies concerning the finite-time blow-up and global existence of solutions to semi-linear pseudo-parabolic equations are well established; however, many challenges remain unresolved. The key to furthering our understanding lies in elucidating the precise effects of pseudo-parabolic terms on key factors such as lifespan, blow-up set, and explosion rate, which are essential for comprehending the asymptotic behavior of these solutions.

This study represents the first attempt to address the challenges associated with the pseudo-parabolic term in Equation (1). Alongside preliminary findings, we have effectively determined the asymptotic behavior and blow-up rate of the solution. Our analysis addresses both low-initial energy and critical-initial energy scenarios, with a strong focus on establishing a lower bound for the maximal existence time.

However, challenges related to the high-initial energy case $\mathrm{E}\left({\mathrm{u}}_0\right)>\mathrm{d}$ remain unaddressed. Specifically, the precise influence of the pseudo-parabolic term on the lifespan, blow-up set, and explosion rate of the pseudo-parabolic equation requires further investigation.