# Blow-Up of Solutions for a Class of Nonlinear Pseudoparabolic Equations with a Memory Term

Authors: Huafei Di; Yadong Shang
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We consider the nonlinear pseudoparabolic equation with a memory termut-Δu-Δut+∫0tλt-τΔuτdτ=div∇up-2u+u1+α, x∈Ω, t>0, with an initial condition and Dirichlet boundary condition. Under negative initial energy and suitable conditions on p, α, and the relaxation function λ(t), we prove a finite-time blow-up result by using the concavity method.


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## Body

## 1. Introduction

In this paper, we consider the initial boundary value problem for a class of nonlinear pseudoparabolic equations with a memory term:(1)ut-βΔu-γΔut+∫0t‍λ(t-τ)Δu(x,τ)dτ=δdiv⁡(|∇u|p-2∇u)+f(u),x∈Ω,t>0,u(x,0)=u0(x),x∈Ω,u(x,t)|∂Ω=0,x∈∂Ω,t>0,
where Ω⊂Rn is a bounded domain with a smooth boundary ∂Ω, λ:R+→R is a given continuous function, β, γ, and δ are all real constant parameters, p>2, and div⁡(|∇u|p-2∇u) is the so-called p-Laplace operator. This type of equations describes a variety of important physical processes, such as the analysis of heat conduction in materials with memory, electric signals in nonlinear telegraph line with nonlinear damping, viscous flow in materials with memory [1], vibration of nonlinear elastic rod with viscosity [2], nonlinear bidirectional shallow water waves [3], and the velocity evolution of ion-acoustic waves in a collisionless plasma when an ion viscosity is invoked [4].Equation (1) includes many important mathematical physics models.In the absence of the memory term, the viscous term, andp-Laplace operator term (γ=δ=0,λ(s)=0),β=1, the model reduces to semilinear parabolic equation:
(2)ut-Δu=f(u),x∈Ω,t>0.
On the existence, nonexistence, and the properties of solutions of (2), there have been many results [5–9].In the absence of the memory term andp-Laplace operator term (δ=0,λ(s)=0),β=γ=1, the model reduces to semilinear pseudoparabolic equation:
(3)ut-Δu-Δut=f(u),x∈Ω,t>0.
Kaikina et al. [10] discussed the periodic boundary value problem of (3) under some assumption forms of nonlinear function f. Cao et al. [11] investigated a class of periodic problems of pseudoparabolic type equations with nonlinear periodic sources. A rather complete classification of the exponent p was given, in terms of the existence and nonexistence of nontrivial and nonnegative periodic solutions. Cao et al. [12] dealt with the Cauchy problem for semilinear pseudoparabolic equations. Existence and uniqueness of local solutions were proved, and the large-time behavior was investigated. Kaikina [13] and Xu and Su [14] discussed the initial boundary value problems of pseudoparabolic equation (3) under some classes of nonlinear function f(u). They obtained some sufficient conditions of existence and uniqueness of local solutions and the large-time behavior of global solutions.In the absence of the memory term and the viscous term(λ(s)=0,γ=0),β=0,δ=1, (1) becomes nonlinear parabolic equation with p-Laplace nonlinear term:
(4)ut=div⁡(|∇u|p-2∇u)+f(u),x∈Ω,t>0.
Tsutsumi [15] studied the initial boundary value problem of (4) with f(u)=u1+α, where p<2+α. He obtained the existence of global weak solutions by using the potential well method. Liu and Zhao [16] considered the same problem with critical initial conditions J(u0)=d or I(u0)=0 and proved the existence of global solution for this problem. Xu et al. [17] discussed (4) at the high energy level, where p<2+α<∞ if n≤p and p<2+α≤np/(n-p) if n>p. They proved the finite time blow-up of solutions by the comparison principle and variational methods. Messaoudi in [18] considers an initial boundary value problem related to (4) and proves, under suitable conditions on f, a blow-up result for solutions with vanishing or negative initial energy.In the absence of the viscous term andp-Laplace operator (γ=0,δ=0),β=0, Gripenberg [19] considered the nonlinear parabolic equation with Volterra integral term equation:
(5)ut=∫0t‍k(t-s)σ(ux(x,s))xds+f(x,t),WWWWWfWWWx∈(0,1),t≥0.
He investigated the initial boundary value problem of (5) and established the global existence of a strong solution of the problem.In the absence of the viscous term andp-Laplace operator (γ=0,δ=0), as β=1, the model reduces to the equation
(6)ut-Δu=∫0t‍b(t-τ)Δu(τ)dτ+f(u),x∈Ω,t>0.
Yin [20] obtained the global existence of a classical solution of (7) under the assumption of a one-sided growth condition. Messaoudi [21] investigated a semilinear parabolic equation with the viscoelastic memory term. He established the finite time blow-up result for the solution with negative or vanishing initial energy for nonlinear function f(u)=|u|p-2u.To the best of our knowledge, there are few works on the study of nonlinear pseudoparabolic equation with memory term of Volterra integral type. Shang and Guo [22–24] investigated the initial boundary value problem and initial value problem of the nonlinear pseudoparabolic equations with Volterra integral term:
(7)ut-f(u)xx-uxxt+∫0t‍λ(t-s)σ(u(x,s),ux(x,s))xds=f(x,t,u,ux),x∈(0,1),t>0.
They proved the existence, uniqueness, and regularities of the global strong solution and gave some conditions of the nonexistence of global solution. In 2007, Ptashnyk [25] investigated the initial boundary value of degenerate quasilinear pseudoparabolic equations with memory term. He obtained some existence results of global solutions. Up to now, there are not any research works on the multidimensional nonlinear pseudoparabolic equations with memory term.In the present work, we deal with the initial boundary problem of the nonlinear pseudoparabolic equation with the memory term of Volterra integral type, the damping term, andp-Laplace operator:
(8)ut-Δu-Δut+∫0t‍λ(t-τ)Δu(τ)dτ=div⁡(|∇u|p-2∇u)+u1+α,x∈Ω,t>0,u(x,0)=u0(x),x∈Ω,u(x,t)|∂Ω=0,x∈∂Ω,t>0,
where Ω⊂Rn is a bounded domain, λ(s):R+→R is a given continuous function, β,γ,δ,>0, and div⁡(|∇u|p-2∇u) is the so-called p-Laplace operator. By using the concavity method first introduced by Levine [5], under negative initial energy and suitable conditions on p,α, and the relaxation function λ(t), we prove that there exists finite-time blow-up solution.Without loss of generality, we chooseβ=γ=δ=1 in the following discussion.


## 2. Preliminaries and Main Results

In this section, we introduce some notations, basic definitions, and important lemmas which will be needed in this paper.For functionsu(x,t),v(x,t) defined on Ω, we introduce
(9)(u,v)=∫Ω‍uvdx,∥u∥2=(∫Ω‍|u|2dx)1/2,∥u∥p=(∫Ω‍|u|pdx)1/p,∥u∥Hm=(∑|α|≤m‍∥Dαu∥22)1/2,∥u∥∞=esssup⁡x∈Ω|u(x)|,Δ=∂2∂x12+⋯+∂2∂xn2,∇=(∂∂x1,…,∂∂xn).We now construct a space of functions as follows. LetHm(Ω) denote the Sobolev space with the norm ∥u∥Hm=(∑|α|≤m‍∥Dαu∥22)1/2. C0∞(Ω) denotes the class of C∞ functions with the compact support in Ω. H0m(Ω) denotes the closure in Hm(Ω) of C0∞(Ω). The Hilbert space H0m(Ω) is a subspace of the Sobolev space Hm(Ω).The following are the basic hypotheses to establish the main results of this paper:(a)
2<p≤2+α<∞;(b)
λ is a C1 function satisfying
(10)λ(τ)≥0,λ′(τ)≤0,(11)∫0∞‍λ(τ)dτ<α(α+2)α(α+2)+1.To obtain the results of this paper, we will introduce the “modified” energy function:(12)E(t)=12(λ∘∇u)(t)+12(1-∫0t‍λ(τ)dτ)∥∇u(t)∥22+1p∥∇u(t)∥pp-12+α∫Ω‍u2+αdx,
where
(13)(λ∘v)(t)=∫0t‍λ(t-τ)∥v(t)-v(τ)∥22dτ.
The following lemma is similar to the lemma of [21] with slight modification.Lemma 1.
Assume that (10) hold. Let p satisfy (a) and let u be a solution of (8). Then E(t) is nonincreasing function; that is
(14)E′(t)≤0.
Moreover, the following energy inequality holds:
(15)E(t)+∫0t‍∥ut(τ)∥H12dτ<E(0).Proof.
By multiplying the equation in (8) by ut, integrating over Ω, we obtain
(16)ddt[∫0t‍∥ut(τ)∥22dτ+∫0t‍∥∇ut(τ)∥22dτ+12∫Ω‍|∇u|2dxhhh+1p∥∇u∥pp-12+α∫Ω‍u2+αdx]-∫0t‍λ(t-τ)∫Ω‍∇ut(t)∇u(τ)dxdτ=0.
For the last term on the left side of (16),
(17)∫0t‍λ(t-τ)∫Ω‍∇ut(t)∇u(τ)dxdτ=∫0t‍λ(t-τ)∫Ω‍∇ut(t)[∇u(τ)-∇u(t)]dxdτ+∫0t‍λ(t-τ)∫Ω‍∇ut(t)∇u(t)dxdτ=-12∫0t‍λ(t-τ)[ddt∫Ω‍|∇u(τ)-∇u(t)|2dx]dτ+12∫0t‍λ(τ)[ddt∫Ω‍|∇u(t)|2dx]dτ=-12ddt[∫0t‍λ(t-τ)∫Ω‍|∇u(τ)-∇u(t)|2dxdτ]+12ddt[∫0t‍λ(τ)∫Ω‍|∇u(t)|2dxdτ]+12∫0t‍λ′(t-τ)∫Ω‍|∇u(τ)-∇u(t)|2dxdτ-12λ(t)∫Ω‍|∇u(t)|2dx.
Inserting (17) into (16), we have
(18)ddt[∫0t‍∥ut(τ)∥22dτ+∫0t‍∥∇ut(τ)∥22dτ+12∫Ω‍|∇u|2dxhhh+1p∥∇u∥pp-12+α∫Ω‍u2+αdx]+12ddt[∫0t‍λ(t-τ)∫Ω‍|∇u(τ)-∇u(t)|2dxdτ]-12ddt[∫0t‍λ(τ)∫Ω‍|∇u(t)|2dxdτ]=12∫0t‍λ′(t-τ)∫Ω‍|∇u(τ)-∇u(t)|2dxdτ-12λ(t)∫Ω‍|∇u(t)|2dx≤0,
for regular solution. The proof of Lemma 1 is completed. This result is valid for weak solutions by a simple density argument.Now we consider the finite time blow-up of solutions withE(0)<0 for the problem (8).Theorem 2.
Letp satisfy (a) and let the relaxation function λ(s) be a C1 function satisfying (10) and (11). Assume that u0∈H01(Ω) such that E(0)<0. Then the solutions u(x,t) of the problem (8) blow up in finite time; that is, the maximum existence time Tmax⁡ of u(x,t) is finite and
(19)lim⁡t↦Tmax⁡∥u(x,t)∥H12=+∞.Proof.
The proof makes use of the so-called “concavity” arguments. For anyT0>0, let
(20)M(t)=∫0t‍∥u(τ)∥22dτ+∫0t‍∥∇u(τ)∥22dτ+(T0-t)(∥∇u0∥22+∥u0∥22),
for t∈[0,T0].
A direct computation yields(21)M′(t)′=∥u(t)∥22+∥∇u(t)∥22M′hh(t)-(∥∇u0∥22+∥u0∥22)M′′(t)=2∫0t‍(u,ut)dτ+2∫0t‍(∇u,∇ut)dτ,M′′(t)=2(u,ut)+2(∇u,∇ut).
By multiplying (8) with u and integrating over Ω,
(22)(u,ut)+(∇u,∇ut)=-∥∇u(t)∥22-∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dx=-∥∇u(t)∥pp+∫Ω‍u2+αdx.
This implies that
(23)M′′(t)=-2∥∇u(t)∥22-2∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dx-2∥∇u(t)∥pp+∫Ω‍u2+αdx,
and we have
(24)M′′(t)M(t)-α+44M′(t)2=2M(t)[-∥∇u(t)∥22-∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dxhhhhhhhhl[-∥∇u(t)∥22-∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dx-∥∇u(t)∥pp+∫Ω‍u2+αdx]-α+44[2∫0t‍(u,ut)dτ+2∫0t‍(∇u,∇ut)dτ]2=2M(t)[-∥∇u(t)∥22-∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dxhhhhhhhhl[-∥∇u(t)∥22-∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dx-∥∇u(t)∥pp+∫Ω‍u2+αdx]+(α+4)×{∫0t‍H(t)-[M(t)-(T0-t)(∥∇u0∥22+∥u0∥22)][∫0t‍(ut,ut)dτ+∫0t‍(∇ut,∇ut)dτ]hhhhh×[∫0t‍(ut,ut)dτ+∫0t‍(∇ut,∇ut)dτ]},
where
(25)H(t)=[∫0t‍(u,u)dτ+∫0t‍(∇u,∇u)dτ]×[∫0t‍(ut,ut)dτ+∫0t‍(∇ut,∇ut)dτ]-[∫0t‍(u,ut)dτ+∫0t‍(∇u,∇ut)dτ]2.
Using Schwartz’s inequality, we have
(26)(∫0t(u,ut)dτ)2≤∫0t∥u∥22dτ∫0t∥ut∥22dτ,(∫0t(∇u,∇ut)dτ)2≤∫0t∥∇u∥22dτ∫0t∥∇ut∥22dτ,∫0t‍(u,ut)dτ∫0t‍(∇u,∇ut)dτ≤12∫0t‍∥u∥22dτ∫0t‍∥∇ut∥22dτ+12∫0t‍∥ut∥22dτ∫0t‍∥∇u∥22dτ.
By (26), we have
(27)H(t)≥0,t∈[0,T0].
Thus,
(28)M′′(t)M(t)-α+44M′(t)2≥M(t)η(t),
where
(29)η(t)=-(α+4)[∫0t‍(ut,ut)dτ+∫0t‍(∇ut,∇ut)dτ]-2∥∇u(t)∥22-2∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dx-2∥∇u(t)∥pp+2∫Ω‍u2+αdx.
For the third term on the left of (29), we have
(30)-∫Ω‍∫0t‍λ(t-τ)Δu(τ)dτu(t)dx=∫0t‍λ(t-τ)∫Ω‍∇u(τ)∇u(t)dxdτ=∫0t‍λ(t-τ)∫Ω‍∇u(t)∇[u(τ)-u(t)]dxdτ+∫0t‍λ(t-τ)∥∇u(t)∥22dτ.
By (29) and (30), we have
(31)η(t)=-(α+4)∫0t‍∥ut∥H12dτ-2(1-∫0t‍λ(t-τ)dτ)∥∇u(t)∥22+2∫0t‍λ(t-τ)∫Ω‍∇u(t)[∇u(τ)-∇u(t)]dxdτ-2∥∇u∥pp+2∫Ω‍u2+αdx≥-(α+4)∫0t‍∥ut∥H12dτ-2(1-∫0t‍λ(t-τ)dτ)∥∇u(t)∥22-2∥∇u∥pp+2∫Ω‍u2+αdx-2[α+22∫0t‍λ(t-τ)∥∇u(τ)-∇u(t)∥22dτhhhhh+12(α+2)∫0t‍λ(t-τ)∥∇u(t)∥22dτ]=-2(α+2)E(t)+(α+2)(1-∫0t‍λ(t-τ)dτ)×∥∇u(t)∥22+2(α+2)p∥∇u∥pp-(α+4)∫0t‍∥ut∥H12dτ-1α+2∫0t‍λ(τ)dτ∥∇u(t)∥22-2(1-∫0t‍λ(t-τ)dτ)∥∇u(t)∥22-2∥∇u∥pp=-2(α+2)E(t)+α(1-∫0t‍λ(t-τ)dτ)∥∇u(t)∥22-1α+2∫0t‍λ(τ)dτ∥∇u(t)∥22-(α+4)∫0t‍∥ut∥H12dτ+2α-2p+4p∥∇u∥pp.
Using Lemma 1, we have
(32)E(t)+∫0t‍∥ut∥H12dτ<E(0),
and then
(33)η(t)≥-2(α+2)E(0)+α∫0t‍∥ut∥H12dτ+[α(1-∫0t‍λ(τ)dτ)-1α+2∫0t‍λ(τ)dτ]∥∇u(t)∥22=-2(α+2)E(0)+α∫0t‍∥ut∥H12dτ+[α-α(α+2)+1α+2∫0t‍λ(τ)dτ]∥∇u(t)∥22.
Since
(34)∫0∞‍λ(τ)dτ<α(α+2)α(α+2)+1,
this implies that
(35)η(t)>δ,0≤t<∞,
where δ is a positive constant.
From the discussion above, we see that(36)M′′(t)M(t)-α+44M′(t)2≥M(t)δ.
From the definition of M(t), there exists ρ>0, such that
(37)M(t)≥ρ,fort∈[0,T),
and we have
(38)M′′(t)M(t)-α+44M′(t)2≥ρδ.
Thus,
(39)(M(t)-α/4)′′=(-α4)M(t)-α/4-2[M′′(t)M(t)-α+44M′(t)2]≤(-α4)ρδM(T)-(α+8)/4<0.
Hence, this proves that M(t)-α/4 reaches 0 in finite time as t→T1-. Since T1 is independent of T, we may assume that T1<T.
This means(40)lim⁡t→T1-M(t)=+∞
or
(41)lim⁡t→T1-(∫0t‍∥u(τ)∥22dτ+∫0t‍∥∇u(τ)∥22dτhhhhh+(T0-t)(∥∇u0∥22+∥u0∥22)∫0t‍(∫0t‍∥u(τ)∥22dτ+∫0t‍∥∇u(τ)∥22dτ)=+∞.
This implies that
(42)lim⁡t→T1-∫0t‍∥u(τ)∥H12dτ=+∞.
Then, the desired assertion immediately follows.
Remark
1. In the absence of the viscous term (Δut) or p-Laplace operator (div⁡(|∇u|p-2∇u)) for the problem (8), the equation reduced to(1)
ut-Δu+∫0t‍λ(t-τ)Δu(τ)dτ=u1+α,(2)
ut-Δu-Δut+∫0t‍λ(t-τ)Δu(τ)dτ=u1+α,(3)
ut-Δu+∫0t‍λ(t-τ)Δu(τ)dτ=div⁡(|∇u|p-2∇u)+u1+α,and from the process of the proof, we can see that the results of Theorem 2 still hold.



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