Research on the Properties of Solutions to Fourth-Order Pseudo-Parabolic Equations with Nonlocal Sources

Abstract

This paper investigates the initial-boundary value problem for a fourth-order pseudo-parabolic equation with a nonlocal source: $u_t+{\Delta }^{2}u-\Delta u_t={\left|u\right|}^{q-1}u-{\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx}$. By employing the Galerkin method, the potential well method, and the construction of an energy functional, we establish threshold conditions for both the global existence and finite-time blow-up of solutions. Additionally, under the assumption of low initial energy $J\left(u_0\right)<d$, an upper bound for the blow-up time is derived.

1. Introduction

This paper studies the initial-boundary value problem of fourth-order pseudo-parabolic equations:

$$\left\{\begin{array}{ll}{u}_t+{\Delta }^{2}u-\Delta u_t={\left|u\right|}^{q-1}u-{\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx},& \left(x,t\right)\in \Omega \times (0,T),\\ u={\displaystyle \frac{\partial u}{\partial v}}\left(x,t\right)=0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& \left(x,t\right)\in \partial \Omega \times (0,T),\\ u\left(x,0\right)=u_0\left(x\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& x\in \Omega .\end{array}\right.$$

(1)

where $\Omega \subset R^n$ is a bounded domain with a smooth boundary; $\Omega \in {ℝ}^n$ $\left(n\ge 1\right)$ is a bounded domain with $C^2$-boundary. Let $\Omega \subset {ℝ}^n$ $\left(n\ge 1\right)$ be a bounded $C^2$-domain. $u\left(x,0\right)\in H_0^2\left(\Omega \right)$, $u_t\left(x,0\right)\in L^2\left(\Omega \right)$, and boundary conditions ${u|}_{\partial \Omega }={\Delta u|}_{\partial \Omega }==0$, $T\in (0,\infty ]$, $q>1$, $\nu$ denotes the unit outer normal vector on $\partial \Omega$. Assume the zero-mass condition ${\displaystyle {\int }_{\Omega }udx=}{\displaystyle {\int }_{\Omega }{u}_{0}dx}=0$ holds, fulfilling compatibility conditions.

$1<q<{\displaystyle \frac{n+4}{n-4}}$ when $n>4$, $\nu$ denotes the unit outer normal vector on $\partial \Omega$, $\Omega \subset {ℝ}^n$ $\left(n\ge 1\right)$ is a bounded domain with $C^2$-boundary, and the initial data satisfies $u\left(x,0\right)\in H_0^2\left(\Omega \right)$ with zero-mass condition ${\displaystyle {\int }_{\Omega }{u}_{0}dx}=0$, fulfilling compatibility conditions. $T\in (0,\infty ]$. In the problem, the nonlocal source term ${\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx}$ represents the spatial average of the solution $u\left(x,t\right)$ over domain $\Omega$.

Payne and Sattinger [1,2] pioneered the potential well method, which characterizes the global existence and nonexistence of solutions under different potential well conditions. Liu et al. [3,4] refined this approach by introducing a family of potential wells, deriving sharp threshold results for global existence and blow-up. Moreover, they established the vacuum isolation phenomenon of solutions and proved their global existence under critical initial conditions. Qu et al. [5] investigated the initial-boundary value problem for a fourth-order parabolic equation with nonstandard growth sources:

$$\left\{\begin{array}{ll}{u}_t+{\Delta }^{2}u=u^{p\left(x\right)},& \left(x,t\right)\in \Omega \times (0,T),\\ u={\Delta }^{2}u=0,& \left(x,t\right)\in \partial \Omega \times (0,T),\\ u\left(x,0\right)=u_0\left(x\right),\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}& x\in \Omega ,\end{array}\right.$$

with bounded domain $\Omega \subset R^n$ $\Omega \in {ℝ}^n$ $\Omega \subset {ℝ}^n$. By constructing a family of potential wells and employing the concavity method, they established sufficient conditions for the finite-time blow-up of weak solutions and derived explicit estimates for the blow-up time.

The pseudo-parabolic equation

$$u_t-u_{xxt}=F(x,t,u_x,u_{xx})$$

has been extensively studied by numerous scholars [6,7,8]. Xu and Su [7] examined the case where $F(x,t,u_x,u_{xx})=u_{xx}+u^p$, establishing the existence and asymptotic behavior of solutions under both subcritical and critical initial energy conditions. Xu and Zhou [9] further examined the semi-linear pseudo-parabolic system with Dirichlet boundary conditions:

$$\left\{\begin{array}{ll}{u}_t=\Delta u_t+\Delta u+{\left|u\right|}^{p-1}u,& (x,t)\in \Omega \times (0,T),\\ u(x,t)=0,& (x,t)\in \partial \Omega \times (0,T),\\ u(x,0)=u_0(x),\hspace{1em}& x\in \Omega ,\end{array}\right.$$

where $\Omega \subset R^n$ $\Omega \in {ℝ}^n$ $\Omega \subset {ℝ}^n$ is a bounded domain with smooth boundary $\partial \Omega \left(n\ge 1\right)$, $p>1$, and $u_0\in H_0^1\left(\Omega \right)$. The authors derived a refined blow-up criterion: when the initial energy satisfies

$$J\left(u_0\right)<{\Lambda }_{u_0}:={\displaystyle \frac{{\lambda }_1\left(p-1\right)}{2\left(p+1\right)\left(1+{\lambda }_1\right)}}{‖u_0‖}_{H_0^1}^2$$

the solution blows up at finite time $T$. The upper bound for the blow-up time is

$$T\le T_{U2}:={\displaystyle \frac{8\left(p+1\right)\left({\lambda }_1+1\right){‖u_0‖}_{H_0^1}^2}{{\left(p-1\right)}^2\left[{\lambda }_1\left(p-1\right){‖u_0‖}_{H_0^1}^2-2\left(p+1\right)\left({\lambda }_1+1\right)J\left(u_0\right)\right]}}$$

For fourth-order problems, Wang and Xu [10] studied the semi-linear pseudo-parabolic equation with a nonlocal nonlinearity source

$$u_t-\Delta u-\Delta u_t={\left|u\right|}^{p-1}u-{\displaystyle \frac{1}{\Omega }}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{p-1}udx},$$

under Neumann boundary conditions. For subcritical energy $J(u_0)<d$, they established refined conditions for global existence, uniqueness, and blow-up.

Zangwill [11] formulated the following fourth-order equation from physical processes such as phase transitions and thin film dynamics:

$$u_t+{\Delta }^{2}u-\nabla \cdot (f(\nabla u))=g(x,t,u).$$

where the $f$ satisfies

$$f(\nabla u)=A_1\nabla u+A_2{\left|\nabla u\right|}^2\nabla u+A_3\nabla {\left|\nabla u\right|}^2$$

and g describes Gaussian noise to account for stochastic fluctuations. These equations have significant applications in many important processes.

Qu and Zhou investigated the following thin film equation:

$$u_t+u_{xxxx}={\left|u\right|}^{q-1}u-{\displaystyle \frac{1}{\Omega }}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx}$$

By employing the potential well method, they established the threshold conditions for the global existence and nonexistence of sign-changing weak solutions. Furthermore, they derived criteria for the finite-time extinction of global solutions.

Subsequently, Li, Gao, and Han [12,13] extended this analysis to the following modified equation:

$$u_t+u_{xxxx}-{\left({\left|u_x\right|}^{p-2}{u}_x\right)}_x={\left|u\right|}^{q-1}u-{\displaystyle \frac{1}{\Omega }}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx}$$

For this model, the authors studied the global existence, uniqueness, finite-time blow-up, and asymptotic behavior of solutions under different initial energy conditions.

Inspired by these works, we consider the fourth-order pseudo-parabolic Equation (1) and prove the threshold conditions for both global existence and the finite-time blow-up of solutions under low initial energy, along with an upper bound estimate for the blow-up time.

2. Preliminaries

This section provides some denotes and Lemmas. Let $L^p\left(\Omega \right)$ $\left(1\le p\le \infty \right)$ be the space of all measurable functions on $\Omega$ satisfying ${\displaystyle {\int }_{\Omega }{\left|u\left(x\right)\right|}^{p}dx<\infty }$, equipped with the norm ${‖\cdot ‖}_p$. For $\left(1\le p\le \infty \right)$ and $m\in N$, the Sobolev space is defined as

$$W^{m,p}\left(\Omega \right)=\left\{u\in L^p\left(\Omega \right)\left|D^{\alpha }u\in L^p\left(\Omega \right),\left|\alpha \le m\right|\right.\right\}$$

$$W^{m,p}\left(\Omega \right)=\left\{u\in L^p\left(\Omega \right)\left|D^{\alpha }u\in L^p\left(\Omega \right),\left|\alpha \right|\le m\right.\right\}$$

where $D^{\alpha }u$ denotes the weak derivative of $u$ of order $\alpha$, with $W_N^{m,p}\left(\Omega \right)=\left\{u\in W^{m,p}\left(\Omega \right)|{\displaystyle \frac{\partial u}{\partial \nu }}{|}_{\partial \Omega }=0,{\displaystyle {\int }_{\Omega }udx=0}\right\}$ denoting the subspace with vanishing normal derivatives on $\partial \Omega$. $H_N^k=W_N^{k,2}\left(k=1,2,\cdots \right)$ for $k=1,2,\cdots$.

Remark 1.

When $m=\infty$, we define $W^{\infty ,p}\left(\Omega \right){\cap }_{m\in N}{W}^{m,p}\left(\Omega \right)$ with the projective limit topology.

The norm $‖\cdot ‖$ on $H^1\left(\Omega \right)$ is

$$‖u‖={‖u‖}_{H^1}={\left({‖u‖}_2^2+{‖\nabla u‖}_2^2\right)}^{{\displaystyle \frac{1}{2}}}.$$

Note that $‖\cdot ‖$ is equivalent to ${‖\nabla \left(\cdot \right)‖}_2$.

Define the energy functional as

$$J\left(u\right)={\displaystyle \frac{1}{2}}{‖\Delta u‖}_2^2-{\displaystyle \frac{1}{q+1}}{‖u‖}_{q+1}^{q+1},$$

(2)

$$I\left(u\right)={‖\Delta u‖}_2^2-{‖u‖}_{q+1}^{q+1}.$$

(3)

The Nehari manifold is given by

$$\mathrm{N}=\left\{u\in H_N^2\left(\Omega \right)\left|I\left(u\right)=0\right.\right\},$$

Further, define the unstable set

$$W=\left\{u\in H_N^2\left(\Omega \right)\left|I\left(u\right)>0,J\left(u\right)<d\right.\right\}\cup \left\{0\right\},$$

$$V=\left\{u\in H_N^2\left(\Omega \right)\left|I\left(u\right)<0,J\left(u\right)<d\right.\right\}.$$

and the depth of potential well

$$d=\underset{u\in \mathrm{N}}{\inf }J\left(u\right).$$

From the definitions of $J\left(u\right)$ and $I\left(u\right)$, it easy to verify that

$$J\left(u\right)={\displaystyle \frac{1}{q+1}}I\left(u\right)+{\displaystyle \frac{q-1}{2\left(q+1\right)}}{‖\Delta u‖}_2^2.$$

(4)

Lemma 1.

Let $u\in H_N^2$; then, the following hold:

  (i)

$\underset{\lambda \to 0^{+}}{\lim }J\left(\lambda u\right)=0;\underset{\lambda \to +\infty }{\lim }J\left(\lambda u\right)=-\infty ;$

 (ii)

There exists a unique ${\lambda }^{\ast }={\lambda }^{\ast }\left(u\right)>0$, such that $J\left(\lambda u\right)$ is strictly increasing for $0<\lambda <{\lambda }^{\ast }$, strictly decreasing for $\lambda <{\lambda }^{\ast }<+\infty$, and attains its maximum at $\lambda ={\lambda }^{\ast }$;

(iii)

For $0<\lambda <{\lambda }^{\ast }$, $I\left(\lambda u\right)>0$; For $\lambda <{\lambda }^{\ast }<+\infty$, $I\left(\lambda u\right)<0$, and $I\left({\lambda }^{\ast }u\right)=0$.

Proof. 

(i) From the definition of $J\left(u\right)$, we have

$$J\left(\lambda u\right)={\displaystyle \frac{{\lambda }^2}{2}}{‖\Delta u‖}_2^2-{\displaystyle \frac{{\lambda }^{q+1}}{q+1}}{‖u‖}_{q+1}^{q+1}.$$

Since $q>1$, it follows that $\underset{\lambda \to 0^{+}}{\lim }J\left(\lambda u\right)=0$ and $\underset{\lambda \to +\infty }{\lim }J\left(\lambda u\right)=-\infty ;$

(ii) [14] Note that

$${\displaystyle \frac{d}{d\lambda }}J\left(\lambda u\right)=\lambda \left({‖\Delta u‖}_2^2-{\lambda }^{q-1}{‖u‖}_{q+1}^{q+1}\right).$$

The critical point ${\lambda }^{\ast }$ is uniquely determined by

$${\lambda }^{\ast }={\left({\displaystyle \frac{{‖\Delta u‖}_2^2}{{‖u‖}_{q+1}^{q+1}}}\right)}^{{\displaystyle \frac{1}{q-1}}}>0,$$

yielding the claimed monotonicity.

(iii) Using the identity

$$I\left(\lambda u\right)={\lambda }^2\left({‖\Delta u‖}_2^2-{\lambda }^{q-1}{‖u‖}_{q+1}^{q+1}\right)=\lambda {\displaystyle \frac{d}{d\lambda }}J\left(\lambda u\right),$$

the result of (ii) holds. □

Definition 1

(Weak Solution). Let $u_0\in H_N^2\left(\Omega \right)$. A function $u\in L^{\infty }\left(0,T;H_N^2\right)$ with $u_t\in L^2\left(0,T;H_N^2\right)$ is called a weak solution of problem (1) if it satisfies

$$\left(u_t,v\right)+\left(\Delta u,\Delta v\right)+\left(\nabla u_t,\nabla v\right)=\left(f\left(u\right),v\right), \forall v\in H_N^2\left(\Omega \right),$$

(5)

and $u\left(x,0\right)=u_0$, where $f={\left|u\right|}^{q-1}u-{\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx}$. Moreover, if (5) holds for every $T>0$, $u\left(x,t\right)$ is called a global (weak) solution.

Energy Identity: Any weak solution satisfies

$${\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^2}d\tau +J\left(u\right)=J\left(u_0\right).$$

(6)

Lemma 2

([15]). $C_{\ast }$ is the optimal embedding constant for $H_N^2\left(\Omega \right)\to L^{q+1}\left(\Omega \right)$, defined as

$$C_{\ast }=\underset{u\in H_N^2}{\sup }{\displaystyle \frac{{‖u‖}_{q+1}}{{‖\Delta u‖}_2}}.$$

Lemma 3.

The depth $d$ of the potential well is given by

$$d={\displaystyle \frac{q-1}{2\left(q+1\right)}}{C}_{\ast }^{-{\displaystyle \frac{2\left(q+1\right)}{q-1}}}>0,$$

Proof. 

For $u\in \mathrm{N}$, we have ${‖\Delta u‖}_2^2={‖u‖}_{q+1}^{q+1}$. Thus,

$$J\left(u\right)={\displaystyle \frac{1}{2}}{‖\Delta u‖}_2^2-{\displaystyle \frac{1}{q+1}}{‖u‖}_{q+1}^{q+1}=\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q+1}}\right){‖\Delta u‖}_2^2$$

therefore, $J\left(u\right)>0$.

By Lemma 1 and Lemma 2, for any $\forall v\in H_N^2\left(\Omega \right)$, there exists a unique ${\lambda }^{\ast }={\left({\displaystyle \frac{{‖\Delta v‖}_2^2}{{‖v‖}_{q+1}^{q+1}}}\right)}^{{\displaystyle \frac{1}{q-1}}}$ such that $u={\lambda }^{\ast }v\in \mathrm{N}$. Consequently,

$$\begin{array}{ll}d=\underset{u\in \mathrm{N}}{\inf }J\left(u\right)& =\underset{u\in \mathrm{N}}{\inf }\left[\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{q+1}}\right){‖\Delta u‖}_2^2\right]\\ & ={\displaystyle \frac{q-1}{2\left(q+1\right)}}\underset{u\in \mathrm{N}}{\inf }{\left({\displaystyle \frac{{‖\Delta u‖}_2^{q+1}}{{‖u‖}_{q+1}^{q+1}}}\right)}^{{\displaystyle \frac{2}{q-1}}}\\ & ={\displaystyle \frac{q-1}{2\left(q+1\right)}}\underset{u\in H_N^2}{\inf }{\left({\displaystyle \frac{{‖\Delta \left({\lambda }^{\ast }v\right)‖}_2^{q+1}}{{‖{\lambda }^{\ast }v‖}_{q+1}^{q+1}}}\right)}^{{\displaystyle \frac{2}{q-1}}}\\ & ={\displaystyle \frac{q-1}{2\left(q+1\right)}}{C}_{\ast }^{-{\displaystyle \frac{2\left(q+1\right)}{q-1}}}\end{array}.$$

□

Lemma 4.

Let $u_0\in H_N^2$, and $u\left(x,t\right)$ be a weak solution of (1).

 (i)

If $u_0\in W$, then $u\left(t\right)\in W$ for all $t\in \left[0,T\right)$.

(ii)

If $u_0\in V$, then $u\left(t\right)\in V$ for all $t\in \left[0,T\right)$, and

$${\displaystyle \frac{q-1}{2\left(q+1\right)}}{‖\Delta u‖}_2^2>d.$$

Proof. 

(i) Since $u_0\in W$, the energy identity (6) yields

$${\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^2}d\tau +J\left(u\right)=J\left(u_0\right)<d.$$

Thus, $J\left(u\right)<d$ for all $t\in \left[0,T\right)$. To show $I\left(u\right)>0$, suppose otherwise. According to the continuity of $I\left(u\right)$ and $I\left(u\left(0\right)\right)>0$, there exists $t\in \left[0,t_0\right)$ such that $I\left(u\right)>0$ for $t\in \left[0,t_0\right)$ and $I\left(u\left(t_0\right)\right)=0$, $\left(u\left(t_0\right)\in \mathrm{N}\right)$. According to the definition of $d$, this implies $J\left(u\left(t_0\right)\right)\ge d$, contradicting $J\left(u\right)<d$. Hence, $u\left(t\right)\in W$ for $t\in \left[0,T\right)$.

(ii) Analogously, if $u_0\in V$, then $u\left(t\right)\in V$ for $t\in \left[0,T\right)$, and $I\left(u\right)<0$; this means ${‖\Delta u‖}_2^2<{‖u‖}_{q+1}^{q+1}$. According to Lemma 1 (iii), there exists ${\lambda }^{\ast }={\left({\displaystyle \frac{{‖\Delta u‖}_2^2}{{‖u‖}_{q+1}^{q+1}}}\right)}^{{\displaystyle \frac{1}{q-1}}}\in \left(0,1\right)$ such that $I\left({\lambda }^{\ast }u\right)=0$, ${\lambda }^{\ast }u\in \mathrm{N}$. Combining (4) and the definitions of $d$, we obtain

$${\displaystyle \frac{q-1}{2\left(q+1\right)}}{‖\Delta u‖}_2^2>{\displaystyle \frac{q-1}{2\left(q+1\right)}}{‖{\lambda }^{\ast }\Delta u‖}_2^2+{\displaystyle \frac{1}{q+1}}I\left({\lambda }^{\ast }u\right)=J\left({\lambda }^{\ast }u\right)\ge d.$$

□

Lemma 5

([13]). Let $\phi \left(t\right)$ be a twice-differentiable function satisfying the following inequality:

$${\phi }^{″}\left(t\right)\phi \left(t\right)-\left(1+\theta \right){\left({\phi }^{\prime }\left(t\right)\right)}^2\ge 0, \forall t\ge 0,$$

where $\theta >0$. If $\phi \left(0\right)>0$ and ${\phi }^{\prime }\left(0\right)>0$; then, $\phi \left(t\right)$ blows up at a finite time $T$, with the upper boun: $T\le {\displaystyle \frac{\phi \left(0\right)}{\theta {\phi }^{\prime }\left(0\right)}}<+\infty$.

Definition 2.

Let $u\left(x,t\right)$ be a weak solution of Problem (1). The maximal existence time $T$ is defined as follows:

• If $u\left(x,t\right)$ exists for $t\in \left(0,T\right)$, but ceases to exist at $t=T$, then $T$ is finite.
• If $u\left(x,t\right)$ exists for $t\in \left[0,\infty \right)$, then $T=+\infty$.

Definition 3

(Finite time blow-up). A weak solution of problem (1) is said to blow up in finite time if there exists $T<+\infty$, such that $\underset{t\to T}{\lim }\left({\displaystyle {\int }_0^t{‖u‖}_{H^1}^{2}d\tau }\right)=+\infty$.

3. Global Existence and Blow-Up for Low Initial Energy $J(u_0)<d$

In this section, we establish the global existence, uniqueness, and finite-time blow-up of solutions under the condition $J(u_0)<d$.

3.1. The Global Existence and Uniqueness of the Solution

Theorem 1.

Let $u_0\left(x\right)\in H_N^2\left(\Omega \right)$ with $I\left(u\right)>0$  $I\left(u_0\right)>0$ and $J\left(u\right)<d$ $J\left(u_0\right)<d$. Then, problem (1) admits a unique global weak solution $u\in L^{\infty }\left(0;T;H_N^2\right)$ with $u_t\in L^2\left(0,T;H_N^2\right)$.

Proof. 

Let ${\left\{{\omega }_j\left(x\right)\right\}}_{j=1}^{\infty }$ be an orthonormal basis of $H_N^2\left(\Omega \right)$. Construct approximate solutions:

$$u_m\left(x,t\right)={\displaystyle \sum _{j=1}^m{g}_{jm}\left(t\right)}{\omega }_j\left(x\right), m=1,2,\cdots ,$$

satisfying the Galerkin system:

$$\left(u_{mt},{\omega }_j\right)+\left(\Delta u_m,\Delta {\omega }_j\right)+\left(\nabla u_t,\nabla {\omega }_j\right)=\left(f\left(u_m\right),{\omega }_j\right), m=1,2,\cdots ,$$

(7)

with initial data

$$u_m\left(x,0\right)={\displaystyle \sum _{j=1}^m{g}_{jm}\left(0\right)}{\omega }_j\left(x\right)\to u_0\left(x\right), \mathrm{in} H_N^2\left(\Omega \right).$$

(8)

where $f\left(u_m\right)={\left|u_m\right|}^{q-1}{u}_m-{\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u_m\right|}^{q-1}{u}_{m}dx}$. The existence of a local solution to Problems (7)–(8) can be obtained using Peano’s Theorem.

Equations (7) and (8) provide an initial value problem for a system of ordinary differential equations:

$$\left\{\begin{array}{c}\left(1+{\lambda }_j\right){\displaystyle \frac{d}{dt}}{g}_{jm}\left(t\right)=f^{\ast }\left(g_{1m},g_{2m},\cdots ,g_{mm}\left(t\right),u\left(t\right)\right),\\ g_{jm}\left(0\right)=\left(u_0,{\omega }_j\right)\end{array}\right.$$

(9)

Since $\left(f\left(u\right),{\omega }_j\right)=\left({\left|u\right|}^{q-1}u,{\omega }_j\right)-\left({\displaystyle \frac{1}{\left|\Omega \right|}}{\displaystyle {\int }_{\Omega }{\left|u\right|}^{q-1}udx},{\omega }_j\right)\in C\left(0,T\right)$ and $g_{jm}\left(t\right)\in C^1\left(0,T\right)$ for all $j$, the initial value problem (9) obtains a local solution via standard existence theory for ordinary differential equations. Multiply (7) by ${\displaystyle \frac{d}{dt}}{g}_{jm}\left(t\right)$, sum over $j$, and integrate from 0 to $t$ to derive the energy identity:

$${\displaystyle {\int }_0^t{‖u_{m\tau }‖}_{H^1}^2}d\tau +J\left(u_m\right)=J\left(u_m\left(0\right)\right).$$

Since $u_m\left(x,0\right)\to u_0\left(x\right)\left(m\to \infty \right)$ in $H_N^2\left(\Omega \right)$ and $u_0\in W$, we have

$$J\left(u_m\left(x,0\right)\right)\to J\left(u_0\left(x\right)\right)<d, I\left(u_m\left(x,0\right)\right)\to I\left(u_0\left(x\right)\right)>0, 0\le t<\infty$$

Therefore, for a sufficiently large $m$,

$${\displaystyle {\int }_0^t{‖u_{m\tau }‖}_{H^1}^2}d\tau +J\left(u_m\right)=J\left(u_m\left(0\right)\right)<d, I\left(u_m\left(x,0\right)\right)>0, 0\le t<\infty$$

(10)

which implies that $u_m\left(x,0\right)\in W$. According to Lemma 4 (i), $u_m\left(x,t\right)\in W$. Thus,

$$I\left(u_m\left(x,t\right)\right)>0$$

and

$$J\left(u_m\right)={\displaystyle \frac{q-1}{2\left(q+1\right)}}{‖\Delta u_m‖}_2^2+{\displaystyle \frac{1}{q+1}}I\left(u_m\right)>{\displaystyle \frac{q-1}{2\left(q+1\right)}}{‖\Delta u_m‖}_2^2, 0\le t<\infty$$

Combining with (10), this yields uniform bounds:

$${\displaystyle {\int }_0^t{‖u_{m\tau }‖}_{H^1}^2}d\tau +{\displaystyle \frac{q-1}{2\left(q+1\right)}}{‖\Delta u_m‖}_2^2<d, 0\le t<\infty$$

which implies

$${‖\Delta u_m‖}_2^2\le {\displaystyle \frac{2d\left(q+1\right)}{q-1}} \mathrm{and} {\displaystyle {\int }_0^t{‖u_{m\tau }‖}_{H^1}^2}d\tau <d, 0\le t<\infty$$

For $\gamma ={\displaystyle \frac{q+1}{q}}$ $\gamma ={\displaystyle \frac{q+1}{q}}>1$. According to Lemma 3,

$${‖u_m^q‖}_{\gamma }^{\gamma }={‖u_m‖}_{q+1}^{q+1}\le C_{\ast }^{q+1}{‖\Delta u_m‖}_2^{q+1}\le C_{\ast }^{q+1}{\left({\displaystyle \frac{2\left(q+1\right)}{q-1}}d\right)}^{{\displaystyle \frac{q+1}{2}}}, 0\le t<\infty$$

According to the convergence control theorem [16], there exist a subsequence of $\left\{u_m\right\}$ and a function $u\in L^{\infty }\left(0;T;H_N^2\right)$ and $u_t\in L^2\left(\left(0,T\right)\times \Omega \right)$, such that, as $m\to \infty$,

$$u_m\to u \mathrm{weakly} \ast \mathrm{in} L^{\infty }\left(0;T;H_N^2\left(\Omega \right)\right),$$

$$\Delta u_m\to \Delta u \mathrm{weakly} \mathrm{in} L^{\infty }\left(0;T;H_N^2\left(\Omega \right)\right),$$

$$u_{mt}\to u_t \mathrm{weakly} \mathrm{in} L^2\left(0;\infty ;L^2\left(\Omega \right)\right),$$

$${\left|u_m\right|}^{q-1}{u}_m\to {\left|u\right|}^{q-1}u \mathrm{weakly} \mathrm{in} L^{{\displaystyle \frac{q+1}{q}}}\left(\left(0,\infty \right)\times \Omega \right).$$

For any $j$, letting $m\to \infty$ we have

$$\left(u_t,{\omega }_j\right)+\left(\Delta u,\Delta {\omega }_j\right)+\left(\nabla u_t,\nabla {\omega }_j\right)=\left(f\left(u\right),{\omega }_j\right).$$

Then, for $\left\{{\omega }_{j\left(x\right)}\right\}\left(j=1,2,\cdots \right)\in H_N^2\left(\Omega \right)$, according to the function $v\in H_N^2\left(\Omega \right)$ and $0\le t<\infty$, we can obtain

$$\left(u_t,v\right)+\left(\Delta u,\Delta v\right)+\left(\nabla u_t,\nabla v\right)=\left(f\left(u\right),v\right).$$

Thus, $u$ is a global weak solution of Problem (1). □

Uniqueness. Let $u_1$, $u_2$ be two weak solutions with identical initial data. $u_t\in L^{\infty }\left(0,T,L^r\left(\Omega \right)\right)$, with ($r>n/2$). For any test function $\phi \in H_N^2\left(\Omega \right)$, the weak formulations satisfy

$$\left(u_{1t},\phi \right)+\left(\Delta u_1,\Delta \phi \right)+\left(\nabla u_{1t},\nabla \phi \right)=\left(f\left(u_1\right),\phi \right),$$

$$\left(u_{2t},\phi \right)+\left(\Delta u_2,\Delta \phi \right)+\left(\nabla u_{2t},\nabla \phi \right)=\left(f\left(u_2\right),\phi \right).$$

Taking $\phi =u_1-u_2\in H_N^2\left(\Omega \right)$ and integrating over $\left(0,t\right)$ for any $t>0$, we have

$${\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^t\frac{d}{d\tau }{‖\phi ‖}_{H^1}^2}d\tau +{{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }\left(\Delta \phi \right)}}}^{2}dxd\tau ={\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }\left({\left|u_1\right|}^{q-1}{u}_1-{\left|u_2\right|}^{q-1}{u}_2\right)}}\phi dxd\tau .$$

(11)

Using mean value theorem, we have

$${\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }\left({\left|u_1\right|}^{q-1}{u}_1-{\left|u_2\right|}^{q-1}{u}_2\right)}}\phi dxd\tau \le {{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }q{\left(\left|u_1\right|+\left|u_2\right|\right)}^{q-1}{\left(u_1-u_2\right)}^{2}dxd\tau \le C{\displaystyle {\int }_0^t‖\phi ‖}}}}_2^{2}d\tau .$$

This can be substituted into (11),

$${\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^t\frac{d}{d\tau }{‖\phi ‖}_{H^1}^2}d\tau +{{\displaystyle {\int }_0^t{\displaystyle {\int }_{\Omega }\left(\Delta \phi \left(x,t\right)\right)}}}^{2}dxd\tau \le C{{\displaystyle {\int }_0^t‖\phi ‖}}_2^{2}d\tau .$$

Because of $\phi \left(x,0\right)=0$,

$${‖\phi \left(x,t\right)‖}_{H^1}^2\le C{{\displaystyle {\int }_0^t‖\phi \left(x,t\right)‖}}_2^{2}d\tau \le C{{\displaystyle {\int }_0^t‖\phi \left(x,t\right)‖}}_{H^1}^{2}d\tau .$$

Applying Gronwall’s inequality, ${‖\phi ‖}_{H^1}^2\le 0$, ${‖u_1-u_2‖}_{H^1}^2=0$, which means $u_1=u_2$ a.e. in $\left(\left(0,\infty \right)\times \Omega \right)$.

3.2. Finite Time Blow-Up of Solutions

Let us start by presenting a lemma.

Lemma 6.

Let $J\left(u_0\right)<d$ and assume the initial energy satisfies ${‖u_0‖}_{H^1}^2<\left(q-1\right)a{b}^2$, where the parameters $a\in \left(0,d-J(u_0)\right)$ and $b>{\displaystyle \frac{{‖u_0‖}_{H^1}^2}{(q-1)a}}$ are positive constants.

Theorem 2.

Let $u_0\left(x\right)\in H_N^2\left(\Omega \right)$ satisfy $J\left(u_0\right)<d$ and $I\left(u_0\right)<0$. Then, there exists a finite time $T$ such that $u$ blows up at $T$, i.e.,

$$\underset{t\to T}{\lim }\left({\displaystyle {\int }_0^t{‖u‖}_{H^1}^{2}d\tau }\right)=+\infty .$$

Moreover, the following holds:

$$T\le g_2(b^{\ast })={\displaystyle \frac{4{‖u_0‖}_{H^1}^2}{{(q-1)}^2(d-J(u_0))}}$$

Theorem 3.

$u$ is a weak solution of (1) with $u_0\left(x\right)\in H_N^2\left(\Omega \right)$. If $J\left(u_0\right)<d$ and $I\left(u_0\right)<0$. Then, there exists a finite time $T$ such that $u$ blows up at $T$; that is,

$$\underset{t\to T}{\lim }\left({\displaystyle {\int }_0^t{‖u‖}_{H^1}^{2}d\tau }\right)=+\infty .$$

And the blow-up time is

$$T\le g_2(b^{\ast })={\displaystyle \frac{4{‖u_0‖}_{H^1}^2}{{(q-1)}^2(d-J(u_0))}}$$

Proof. 

For a contradiction, assume that $u$ exists globally. For fixed $\stackrel{\sim }{T}>0$, define

$$M(t):={\displaystyle {\int }_0^t{‖u‖}_{H^1}^{2}d\tau }+(\stackrel{\sim }{T}-t){‖u_0‖}_{H^1}^2+a{(t+b)}^2,0<t<\stackrel{\sim }{T},$$

where $a,b>0$ are determined in the following. Then,

$$M^{\prime }(t)={‖u‖}_{H^1}^2-{‖u_0‖}_{H^1}^2+2a(t+b)={\displaystyle {\int }_0^t\frac{d}{d\tau }{‖u‖}_{H^1}^{2}d\tau }+2a(t+b).$$

Testing (5) with $v=u$ yields

$${\displaystyle \frac{1}{2}}{\displaystyle \frac{d}{dt}}{‖u‖}_{H^1}^2=-\left({‖\Delta u‖}_2^2-{‖u‖}_{q+1}^{q+1}\right).$$

According to the definition of $J\left(u\right)$ and $I\left(u\right)$, (3-2), (3-3),

$$\begin{array}{ll}{M}^{″}(t)& ={\displaystyle \frac{d}{dt}}{‖u‖}_{H^1}^2+2a=-\left({‖\Delta u‖}_2^2-{‖u‖}_{q+1}^{q+1}\right)+2a=-2I(u)+2a\\ & =(q-1){‖\Delta u‖}_2^2-2(q+1)J(u)+2a\end{array}.$$

According to Lemma 4 (ii) and energy identity (6), this becomes

$$\begin{array}{ll}{M}^{″}\left(t\right)& >2d(q+1)+2(q+1)({\displaystyle {\int }_0^t{‖u_{\tau }‖}_H^{2}d\tau })-J(u_0))+2a\\ & >2(q+1)\left((d-J(u_0)+{\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^{2}d\tau }\right)\end{array}.$$

The Cauchy–Schwarz inequality gives

$${\displaystyle {\int }_0^t\frac{d}{dt}{‖u_{\tau }‖}_{H^1}^{2}dt=2{\displaystyle {\int }_0^t{(u,u_{\tau })}_{H^1}d\tau \le 2}}{\left({\displaystyle {\int }_0^t{‖u‖}_{H^1}^{2}d\tau }\right)}^{{\displaystyle \frac{1}{2}}}{\left({\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^{2}d\tau }\right)}^{{\displaystyle \frac{1}{2}}}$$

Thus,

$$\begin{array}{ll}{\left(M^{\prime }(t)\right)}^2& =4{\left({\displaystyle \frac{1}{2}}{\displaystyle {\int }_0^t\frac{d}{dt}{‖u‖}_{H^1}^{2}dt}+a(t+b)\right)}^2\\ & \le 4\left({\displaystyle {\int }_0^t{‖u‖}_{H^1}^{2}d\tau }+a{(t+b)}^2\right)\left({\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^{2}d\tau }+a\right)\\ & \le 4M(t)\left({\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^{2}d\tau }+a\right).\end{array}$$

Combining this with $M^{″}(t)$, we obtain

$$\begin{array}{l}{M}^{″}(t)M(t)-{\displaystyle \frac{q+1}{2}}{\left(M^{\prime }(t)\right)}^2\\ >2(q+1)M(t)\left((d-J(u_0)+{\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^{2}d\tau }\right)-2(q+1)M(t)\left({\displaystyle {\int }_0^t{‖u_{\tau }‖}_{H^1}^{2}d\tau }+a\right)\\ =2(q+1)M(t)(d-J(u_0)-a)\end{array}$$

Choosing $a\in \left(0,d-J(u_0)\right),$ we obtain

$$M^{″}(t)M(t)-{\displaystyle \frac{q+1}{2}}{\left(M^{\prime }(t)\right)}^2>0.$$

For $M(0)=T{‖u_0‖}_{H^1}^2+a{b}^2$, $M^{\prime }(0)=2ab>0$, according to Lemma 5, $M\left(t\right)$ is blow-up in finite time $T$ with

$$T\le {\displaystyle \frac{2M(0)}{(q-1)M^{\prime }(0)}}={\displaystyle \frac{b}{q-1}}+{\displaystyle \frac{{‖u_0‖}_{H^1}^2}{(q-1)ab}}\stackrel{\sim }{T}$$

Setting $\stackrel{\sim }{T}=T$ and choosing $b\in \left({\displaystyle \frac{{‖u_0‖}_{H^1}^2}{(q-1)a}},+\infty \right)$, we arrive at

$$T\le {\displaystyle \frac{a{b}^2}{(q-1)ab-{‖u_0‖}_{H^1}^2}}.$$

(12)

Letting

$$c:=ab\in \left({\displaystyle \frac{{‖u_0‖}_{H^1}^2}{(q-1)a}},(d-J(u_0))b\right),$$

(11) can be rewritten as follows:

$$T\le {\displaystyle \frac{bc}{(q-1)c-{‖u_0‖}_{H^1}^2}}=:g_1(b,c)$$

(13)

Since the function $g_1\left(b,c\right)$ is continuously and monotonically decreasing with respect to $c$, we have

$$\underset{c}{\inf } g_1(b,c)=g_1(b,(d-J(u_0))b)={\displaystyle \frac{(d-J(u_0))b^2}{(q-1)(d-J(u_0))b-{‖u_0‖}_{H^1}^2}}:=g_2(b).$$

$g_2\left(b\right)$ takes the minimum value at $b^{\ast }={\displaystyle \frac{2{‖u_0‖}_{H^1}^2}{(q-1)(d-J(u_0))}}$; From (11), we can see that

$$T\le g_2(b^{\ast })={\displaystyle \frac{4{‖u_0‖}_{H^1}^2}{{(q-1)}^2(d-J(u_0))}}.$$

□

4. Perspectives

The existence theory for solutions of nonlinear parabolic equations constitutes a fundamental research area in partial differential equations, with profound theoretical implications and significant practical applications. The blow-up phenomenon of solutions corresponds to mathematical models describing diverse natural phenomena, including combustion-induced explosions in solids, nuclear reactor dynamics, and shock wave formation in mechanical systems. In this work, we systematically investigate the critical conditions for solution blow-up. While the paper focus on equations with constant exponent $q$, an important open question remains: Can the proposed methodology be extended to analyze fourth-order parabolic equations with variable exponents when $q$ becomes a function of the solution or a spatial–temporal variable? This extension could substantially broaden the applicability of our framework to more realistic physical scenarios where material properties exhibit spatial or temporal heterogeneity.

The existence of solutions to nonlinear parabolic equations is a key problem in the study of partial differential equations. In this paper, we investigate the threshold conditions for an initial-boundary value problem of a fourth-order pseudo-parabolic equation with a nonlocal source. Additionally, under the assumption of low initial energy $J\left(u_0\right)<d$, we provide an upper bound for the blow-up time. There are some open questions—for example, can the proposed methodology be extended to analyze fourth-order with variable exponents when $q$ becomes a function of the solution or spatial-temporal variables? This extension could substantially broaden the applicability of our framework to more realistic physical scenarios where the material properties exhibit spatial or temporal heterogeneity.