Blow-up analysis of a nonlinear pseudo-parabolic equation with memory term

1.   Introduction

Studied here is an initial boundary value problem for the nonlinear pseudo-parabolic equation with memory term

where the exponent $p > 0$, and $\Omega$ with smooth boundary $\partial\Omega$ is a bounded domain in $\mathbb{R}^{n}\; (n\geq1)$. The relaxation function $g$ represents the kernel of memory term, satisfying certain conditions to be specified later. The reason why this type of equations is so attractive is that it has extensive physical background, which appears in the study of heat conduction and viscous flow in materials with memory, electric signals in telegraph line with nonlinear damping ^[1], vibration of nonlinear viscoelastic rod ^[2], bidirectional nonlinear shallow water waves ^[3], and the velocity evolution of ion-acoustic waves in a collision less plasma ^[4] and so on.

The Eq. (1.1) describes many important and famous mathematical models which have extensive theoretical connotation. For example, in the absence of the memory term (i.e., $g\equiv0$), the Eq. (1.1) becomes a semilinear pseudo-parabolic equation

As is known to all, the nonlinear pseudo-parabolic equations appear in the dynamics of the thermodynamics, hydrodynamics and filtration theory, etc (see ^[1,5,6]). There is a lot of literature on the study of such kinds of semilinear pseudo-parabolic equation, such as the existence and uniqueness of certain solutions in ^[1,7,8], blow-up results in ^[8,9,10], asymptotic behavior in ^[8,11] and so on. In ^[8], Xu etc. derived the finite time blow-up results in $H_{0}^{1}(\Omega)$-norm. Luo ^[9] derived the upper bound and lower bound for the blow-up time under some relative assumptions. In recent years, the qualitative properties of parabolic and pseudo-parabolic equations (systems) with nonlinear gradient terms have been attracting some authors' attention and many interesting results have also been obtained. For example, Dong and Zhou ^[12] considered a class of parabolic system with different gradient coefficients and coupled nonlocal sources, of which the global existence and a series of finite time blow-up results were discussed by using comparison principle and asymptotic analysis methods. In ^[13], Marras etc. gave some estimates about the upper and lower bounds of blow-up time for the pseudo-parabolic equation (system) with nonlinear gradient terms under suitable conditions. More results on these types of equations (systems) can be seen in ^[1,12,13,27] and the references therein.

In ^[14], Gripenberg investigated the initial boundary value problem for a volterra integro-differential parabolic equation

He obtained the global existence of a strong solution under certain conditions.

In the absence of dispersion term $\triangle u_{t}$, the model (1.1) is reduced to the following nonlinear parabolic equation

Yin ^[15] discussed the initial boundary value problem of Eq. (1.6) and obtained the global existence of a classical solution under a one-sided growth condition. In ^[16], Messaoudi replaced the memory term $b(t-\tau)$ by $-g(t-\tau)$ in Eq. (1.6). He proved the finite time blow-up of the solutions with negative and vanishing initial energy.

As far as we know, there are few results of viscoelastic pseudo-parabolic equations. In ^[17,18], Shang and Guo studied the initial boundary value problem and initial value problem for a class of integro-differential equations of pseudo-parabolic type. They proved the global existence, uniqueness, asymptotic behavior of solutions for the problem, and gave the sufficient conditions for the nonexistence of global solutions in one dimension case. Ptashnyk ^[19] studied the initial boundary value problem of degenerate viscoelastic quasilinear pseudo-parabolic equations. For the initial value problem of a class of degenerate viscoelastic pseudo-parabolic equations, Carroll and Showalter ^[20] represented some background in physics and applied sciences, and introduced a number of computing techniques for the study of the pseudo-parabolic equations with different types of memory terms. Di and Shang ^[21] considered the initial boundary value problem of Eq. (1.1). They obtained the global existence and finite time blow-up of the solutions with the low initial energy ${E}(0) < d_{k}$ by using the potential well method. Later, the results of ^[21] were improved by Sun, Liu and Wu ^[22], to certain solutions with the low initial energy $J(u_{0}) < d_{\infty}$. Here, the functionals $E(t)$, $J(u(t))$ and $d_{k}$ denote the total energy, potential energy and potential well depth associated with problem (1.1)–(1.3), respectively. About the application of the potential well theory to the study of evolution equations, we also refer the reader to see ^{[21,22,23,24,25]} and references therein.

In this paper, we further extend and improve the blow-up results obtained in ^[21,22], the idea of which are based on the lemmas of Li and Tsai ^[26] (up to a appropriate modification). It is worth mentioning that the method of this paper is not directly related to the potential well theory, we shall discuss the blow-up phenomena of the solutions for problem (1.1)–(1.3) under the initial energy $E(0) < 0$, $0\leq E(0) < \frac{\eta}{p}E_{1}$ and $\frac{\eta}{p}E_{1}\leq E(0) < \frac{\|u_{0}\|_{H^{1}}^{2}}{\gamma}$, respectively. Moreover, we also establish the corresponding upper bounds for blow-up time $T^{*}$ at the three different initial energy levels, which depend on the sign and size of initial energy $E(0)$. Finally, a lower bound of blow-up time $T^{*}$ is obtained by applying a differential inequality technique, if blow-up occurs to the initial boundary problem (1.1)–(1.3).

The outline of this paper is as follows: In Section 2, we introduce some notations, functionals and important lemmas to be used throughout this article. Section 3 is devoted to study the finite time blow-up results for the solutions with initial data at non-positive energy level as well as arbitrary positive energy level, and give some upper bounds for blow-up time $T^{*}$ depending on the sign and size of initial energy $E(0)$. In Section 4, a lower bound for the life span $T^{*}$ is derived by means of a differential inequality technique if blow-up does occur.

2.   Preliminaries

Our attention of this section is to give some notations, functionals and important lemmas in order to state the main results of this paper. Throughout the whole paper, the following abbreviations are used for precise statement:

And $(u, v) = \int_{\Omega}uvdx$ for $L^{2}$-inner product will also denotes the notation of duality paring between dual spaces. The Sobolev space $H_{0}^{1}(\Omega)$ will be defined with the inner product $\langle u, v\rangle = \int_{\Omega}uvdx+\int_{\Omega}\nabla u\cdot \nabla vdx$.

The following conditions are the essential hypotheses to obtain the main results of this paper.

$(i)$ $0 < p < \infty$ if $n = 1, 2$ and $0 < p\leq\frac{4}{n-2}$ if $n\geq3$.

$(ii)$ The relaxation function $g$: $[0, \infty)\rightarrow(0, \infty)$ is a $C^{1}$ function satisfying

and

where $0 < \delta\leq\frac{p}{2}$.

We first start with the following existence and uniqueness of local solution for problem (1.1)–(1.3), which can be obtained by using Faedo-Galerkin methods and Contraction Mapping Principle as in ^[1,27,28]. Here, the proof is thus omitted.

Theorem 2.1. Let the conditions $(i)$, (2.1) hold and $u_{0}\in{H}_{0}^{1}(\Omega)$. Then there exists a unique local solution $u$ of problem (1.1)–(1.3) such that

for some $T > 0$.

To obtain the results of this paper, we define the energy functional for the solution $u$ of problem (1.1)–(1.3) in the form

where $(g\diamond\nabla u)(t) = \int_{0}^{t}g(t-\tau)\|\nabla u(t)-\nabla u(\tau)\|_{2}^{2}d\tau$. Furthermore, multiplying Eq. (1.1) by $u_{t}$ and integrating it over $\Omega$, we easily deduce from $(i)$ and (2.1) that

Remark 2.1. In view of (2.1), (2.4) and Sobolev inequality, we discover that

where the function $h(\lambda) = \frac{1}{2}\lambda^{2}-\frac{B^{p+2}}{(p+2)l^{\frac{p+2}{2}}}\lambda^{{p+2}}$, $\lambda = \left(l\|\nabla u\|_{2}^{2}+(g\diamond\nabla u)(t)\right)^{\frac{1}{2}} > 0$, $B$ is the optimal constant satisfying the Sobolev embedding inequality $\|u\|_{p+2}\leq B\|\nabla{u}\|_{2}$. A direct computation gives that $h(\lambda)$ is increasing for $0 < \lambda < \lambda_{1}$, decreasing for $\lambda > \lambda_{1}$ and $\lambda_{1} = \left(\frac{l}{B^{2}}\right)^{\frac{p+2}{2p}}$ is the absolute maximum point of $h(\lambda)$ satisfying

Lemma 2.1. Assume that the conditions $(i)$, (2.1) hold and $u$ is a solution of problem (1.1)–(1.3) with initial data $E(0) < E_{1}$ and $l^{\frac{1}{2}}\|\nabla u_{0}\|_{2} > \lambda_{1}$. Then there exists $\lambda_{2} > \lambda_{1}$ satisfying

for $t > 0$.

Proof. From Remark 2.1, we know that $h(\lambda)$ is increasing for $0 < \lambda < \lambda_{1}$, decreasing for $\lambda > \lambda_{1}$ and $h(\lambda)\rightarrow-\infty$ as $\lambda\rightarrow\infty$. By $E(0) < E_{1}$, we have that there exist $\lambda_{2}'$ and $\lambda_{2}$ such that $\lambda_{1}\in(\lambda_{2}', \lambda_{2})$ and $h(\lambda_{2}') = h(\lambda_{2}) = E(0)$. To prove (2.8), by contradiction we assume that there exists a time $t_{0} > 0$ such that

(1) If $\lambda_{2}' < \left(l\|\nabla u(t_{0})\|_{2}^{2}+(g\diamond\nabla u)(t_{0})\right)^{\frac{1}{2}} < \lambda_{2}$, it is inferred that

which contradicts (2.6).

(2) Assume that $\left(l\|\nabla u(t_{0})\|_{2}^{2}+(g\diamond\nabla u)(t_{0})\right)^{\frac{1}{2}}\leq\lambda_{2}'$ holds. Considering $l^{\frac{1}{2}}\|\nabla u_{0}\|_{2} > \lambda_{1}$, we get from (2.6) that $h(l^{\frac{1}{2}}\|\nabla u_{0}\|_{2}) < E(0) = h(\lambda_{2})$, which implies $l^{\frac{1}{2}}\|\nabla u_{0}\|_{2} > \lambda_{2}$. Thus, applying the continuity of $\left(l\|\nabla u\|_{2}^{2}+(g\diamond\nabla u)(t)\right)^{\frac{1}{2}}$, we know that there exists a $t_{1}\in(0, t_{0})$ such that $\lambda_{2}' < \left(l\|\nabla u(t_{1})\|_{2}^{2}+(g\diamond\nabla u)(t_{1})\right)^{\frac{1}{2}} < \lambda_{2}$. Hence, we have

which also contradicts (2.6).

Next, we will mention an important lemma which is similar to the lemmas of ^[26] with slight modification.

Lemma 2.2. Let $J(t)$ be a nonincreasing function on $[t_{0}, \infty)$, $t_{0}\geq0$, and satisfy the differential inequality

where $\alpha > 0$ and $\beta < 0$. Then there exists a finite positive time $T^{*}$ such that

and the upper bound for $T^{*}$ is estimated by:

where $J(t_{0}) < \min\left\{1, \sqrt{\frac{\alpha}{-\beta}}\right\}$.

3.   Upper bound for blow-up time

In this section, we will give some blow-up results for the solutions with initial energy $E(0) < 0$, $0\leq E(0) < \frac{\eta}{p}E_{1}$ and $\frac{\eta}{p}E_{1}\leq E(0) < \frac{\|u_{0}\|_{H^{1}}^{2}}{\gamma}$, respectively. Furthermore, some upper bounds for blow-up time $T^{*}$ depending on the sign and size of initial energy $E(0)$ are obtained for problem (1.1)–(1.3).

To obtain the blow-up results and upper bounds, we first define functionals

and

where $\delta\in(0, \frac{p}{2}]$, $T_{0}$ is positive constant to be chosen later and then give the following lemma.

Lemma 3.1. Assume that the conditions $(i)$, $(ii)$ hold, $u_{0}\in{H}_{0}^{1}(\Omega)$ and $u$ is a solution of problem (1.1)–(1.3), then we have

where $G(t) = -4(1+\delta)E(0)+\eta[l\|\nabla u\|_{2}^{2}+(g\diamond\nabla u)(t)]$, $\eta = 1+2\delta-\frac{1}{l} > 0$.

Proof. From (3.1), a direct computation yields

and

By (2.4), (2.5) and (3.5), it is found that

From the Young inequality, it is inferred that

Hence, combining (2.2), (3.6) and (3.7), it follows that

where $\eta = 1+2\delta-\frac{1}{l} > 0$. The proof is completed.

Next, we shall state and prove the finite time blow-up results on the solutions of problem (1.1)–(1.3).

Theorem 3.1. Assume that the conditions $(i)$, $(ii)$ hold and $\|u_{0}\|_{H^{1}}^{2} < \frac{1}{T_{0}}$. Furthermore assume that either one of the following three conditions is satisfied:

(1) $E(0) < 0$;

(2) $0\leq E(0) < \frac{\eta}{p}E_{1}$, and $\lambda_{1} < l^{\frac{1}{2}}\|\nabla u_{0}\|_{2}$;

(3) $\frac{\eta}{p}E_{1}\leq E(0) < \frac{\|u_{0}\|_{H^{1}}^{2}}{\gamma}$.

Then, the solutions $u$ of problem (1.1)–(1.3) blow up in finite time, which means that the maximum existence time $T^{*}$ of $u$ is finite and

Moreover, the upper bounds for blow-up time $T^{*}$ can be estimated according to the sign and size of energy $E(0)$:

Case (1): If $E(0) < 0$, then an upper bound of blow-up time $T^{*}$ is given by

Case (2): If $0\leq E(0) < \frac{\eta}{p}E_{1}$, and $\lambda_{1} < l^{\frac{1}{2}}\|\nabla u_{0}\|_{2}$, then an upper bound of blow-up time $T^{*}$ is given by

Case (3): $\frac{\eta}{p}E_{1}\leq E(0) < \frac{\|u_{0}\|_{H^{1}}^{2}}{\gamma}$, then an upper bound of blow-up time $T^{*}$ is given by

where $Z(0) = \|u_{0}\|_{H^{1}}^{2}-\gamma E(0)$, $\gamma = \frac{4(1+\delta)}{D}$ and $D = \frac{\eta l\mu_{1}}{1+\mu_{1}}$; $\mu_{1}$ is the first eigenvalue of operate $-\triangle$ subject to the homogeneous Dirichlet boundary condition.

Proof. We will consider the different cases depending on the sign and size of initial energy $E(0)$.

Case (1): If $E(0)\leq0$, then it follows from (3.6) that

for $t\geq0$. Thus, we get $a'(t) > a'(0) = \|u_{0}\|_{H^{1}}^{2}\geq0$, for $t > 0$. Obviously, by the direct computation, we have from (3.2) that

and

where

Then, by Lemma 3.1, we obtain that

On the other hand,

Applying the Hölder and Cauchy inequalities, we discover that

and

Inserting (3.14)–(3.18) into (3.13), it follows that

Thus, by (3.12), (3.19) and the definition of $G(t)$, there appears the relation

Note that by $a'(t) > a'(0) = \|u_{0}\|_{H^{1}}^{2}$ for $t > 0$ and (3.11), we deduce that $J'(t) < 0$ for $t > 0$ and $J'(0) = 0$. Multiplying (3.20) with $J'(t)$ and integrating it from $0$ to $t$, then we conclude that

where

where $J(0) = \left(T_{0}\|u_{0}\|_{H^{1}}^{2}\right)^{-\delta} > 1$.

Then, by the combination of (3.21)–(3.23) and Lemma 2.2, we can obtain that there exists a finite time $T^{*}$ satisfying $\lim_{t\rightarrow T^{*-}}J(t) = 0$, which also implies that

Furthermore, from Lemma 2.2, it follows that an upper bound of blow-up time $T^{*}$ is given by

Case (2): If $0\leq E(0) < \frac{\eta}{p}E_{1}$, and $l^{\frac{1}{2}}\|\nabla u_{0}\|_{2} > \lambda_{1}$, then utilizing Lemma 2.1, definition of $\lambda_{1}$ and (2.7), we have

where in the last inequality we have used $0 < \delta\leq\frac{p}{2}$. Inserting (3.25) into (3.8), it follows that

Hence, we also have $a'(t) > a'(0) = \|u_{0}\|_{H^{1}}^{2}\geq0$, for $t > 0$.

Then, using the similar arguments to the case (1), we obtain that

Hence, from (3.26) and (3.27), it follows that

Applying the same discussion as in case (1), we also have $J'(t) < 0$ for $t > 0$ and $J'(0) = 0$. Multiplying (3.28) with $J'(t)$ and integrating it from $0$ to $t$, then we discover that

where

Therefore, we can obtain from Lemma 2.2 and (3.29)–(3.31) that there exists a finite time $T^{*}$ such that $\lim_{t\rightarrow T^{*-}}J(t) = 0$, which means that

Furthermore, from Lemma 2.2, we get that an upper bound of blow-up time $T^{*}$ is given by

Case (3): When $\frac{\eta}{p}E_{1}\leq E(0) < \frac{\|u_{0}\|_{H^{1}}^{2}}{\gamma}$, we define the functional

where $\gamma = \frac{4(1+\delta)}{D}$ and $D = \frac{\eta l\mu_{1}}{1+\mu_{1}} > 0$; $\mu_{1}$ is the first eigenvalue of operate $-\triangle$ subject to the homogeneous Dirichlet boundary condition such that $\| \nabla u\|_{2}^{2}\geq\mu_{1}\| u\|_{2}^{2}$. Using (3.33) and Poincaré inequality, we get that

Then, from

and (3.34), it is deduced that

By (3.34) and (3.36), there appear the relation

Hence, we also have $a'(t) > a'(0) = \|u_{0}\|_{H^{1}}^{2}\geq0$, for $t > 0$.

Next, using the same discussion as in case (1), it follows that

By (3.34) and (3.36) again, we also get that

Hence, it follows that

Multiplying (3.39) with $J'(t)$ and integrating it from $0$ to $t$, then there appears the relation

where

Similarly, we can obtain from Lemma 2.2 and (3.40)–(3.42) that there exists a finite time $T^{*}$ such that

Similarly, from Lemma 2.2, we conclude that an upper bound of blow-up time $T^{*}$ is given by

This completes the proof of Theorem 3.1.

4.   Lower bound for blow-up time

Our goal of this section is turned to determine a lower bound for blow-up time $T^{*}$ when blow up occurs to the initial boundary value problem (1.1)–(1.3).

Theorem 4.1. Assume that the conditions $(i)$, $(ii)$ hold. Let $u_{0}\in{H}_{0}^{1}(\Omega)$ and $u$ be a blow-up solution of problem (1.1)–(1.3), then a lower bound for blow-up time $T^{*}$ can be estimated in the form

where $B$ is the optimal constant satisfying the Sobolev embedding inequality $\|u\|_{p+2}\leq B\|\nabla{u}\|_{2}$ and $F(0) = \|u_{0}\|_{H^{1}}^{2}$.

Proof. Let us define the auxiliary function

Differentiating (4.1) with respect to $t$ and integration by parts, then we discover that

Making use of the Young inequality, we have

Inserting (4.3) into (4.2), it follows that

From (2.1), (2.5), (4.4) and the definitions of $E(t)$, $F(t)$, we have

where

Integrating the inequality (4.5) from 0 to $t$, we deduce that

If $u$ blows up in $H_{0}^{1}(\Omega)$-norm, then we establish a lower bound for $T^{*}$ by the form

which thereby completes the proof of Theorem 4.1.

Acknowledgments

This work was initiated while Di was visiting Department of Mathematics, University of Texas at Arlington as a Postdoctoral Researcher during the year 2018–2020, who would like to thank the department for its warm hospitality and support. The work is supported by the NSF of China (11801108, 11701116), the Scientific Program of Guangdong Province (2016A030310262), and the College Scientific Research Project of Guangzhou City (1201630180).

Conflict of interest

The authors declare that there is no conflicts of interest in this paper.