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Research in Applied Mathematics: Vol. 1
Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101258, 9 pages
doi:10.11131/2017/101258

Exponential Growth of Solution for a Class of Reaction Diffusion Equation with Memory and Multiple Nonlinearities

Jian Dang, Qingying Hu, Suxia Xia, and Hongwei Zhang

Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China

Received 8 November 2016; Accepted 19 August 2017

Editor: Zhouhong Li

Copyright © 2017 Jian Dang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the initial boundary value problem of a class of reaction diffusion equation with memory and multiple nonlinearities. We show the exponential growth of solution with Lp-norm using a differential inequality.

1. Introduction

In this paper, we study the following initial boundary value problem for a class of reaction diffusion equation with memory and multiple nonlinearities

u t Δ u + 0 t g ( t s ) Δ u ( s ) d s + | u | k 2 u t = | u | p 2 u , (1.1)

u ( x , t ) = 0 , x Ω , (1.2)

u ( x , 0 ) = u 0 ( x ) , x Ω , (1.3)

where k>2, p>2 are real numbers and Ω is bounded domain in Rn with smooth boundary Ω so that the divergence theorem can be applied. Here, g represents the kernel of memory term and Δ denotes the Laplace operator in Ω.

This type of problem not only is important from the theoretical point of view, but also arises in many physical applications and describes a great deal of models in applied science. It appears in the models of chemical reactions, heat transfer, population dynamics, and so on (see [1] and references therein). One of the most important field of such problems arises in the models of nonlinear viscoelasticity.

In absence of the nonlinear diffusion term |u|k−2ut, and when the function g vanishes identically (i.e. g = 0), then the equation (1.1) can be reduced to the following equation

u t Δ u = | u | p 2 u , (1.4)

A related problem to the equation (1.4) has attracted great attentions in the last two decades, and many results appeared on the existence, blow-up and asymptotic behavior of solutions. It is well known that the nonlinear reaction term |u|p−2u drives the solution of (1.4) to blow up in finite time and the diffusion term is known to yield existence of global solution if the reaction term is removed from the equation [2]. The more general equation

u t div ( | u | m 2 u ) = f ( u ) , (1.5)

has also attracted a great deal of people. The obtained results show that global existence and nonexistence depend roughly on m, the degree of nonlinearity in f, the dimension n, and the size of the initial data. In this regard, see the works of Levine [3], Kalantarov and Ladyzhenskaya [4], Levine et al. [5], Messaoudi [6], Liu et al. [7], Ayouch [8] and references therein. Pucci and Serrin [9] discussed the stability of the following equation

| u t | l 2 u t div ( | u | m 2 u ) = f ( u ) , (1.6)

Levine et al. [5] got the global existence and nonexistence of solution for (1.6). Pang et. al [10,11] and Berrimi [12] gave the sufficient condition of blow-up result for certain solutions of (1.6) with positive or negative initial energy.

When g = 0, the class of equation (1.1) can also be as a special case of doubly nonlinear parabolic-type equations (or the porous medium equation) [13,5]

β ( u ) t Δ u = | u | p 2 u (1.7)

if we take β(u) = u + |u|m−2u. Such equation play an important role in physics and biology. It should be noted that questions of solvability, local and global in time, asymptotic behavior and blow-up of initial boundary value problems and initial value problems for equation of the type (1.7) were investigated by many authors. We only mention the work [13,14] for this class equation.

We should also point out that Polat [15] established a blow-up result for the solution with vanishing initial energy of the following initial boundary value problem

u t u x x + | u | m 2 u t = | u | p 2 u .

They also gave detailed results of the necessary and sufficient blow-up conditions together with blow-up rate estimates for the positive solution of the problem

( u m ) t Δ u = f ( u ) ,

subjected to various boundary conditions. Korpusov [16,17] have been obtained sufficient conditions for the blowup of a finite time and solvability for the following generalized Boussinesq equation

u t Δ u Δ u t + | u | m 2 u t = u ( u α ) ( u β ) , (1.8)

with initial boundary value (1.2) and (1.3) in R3 for α, β>0 by concavity method [3,4]. The result is extended by recent paper [18,19,20,21].

In absence of the nonlinear diffusion term |u|k−2ut, and with the presence of the viscoelastic term (i.e. g≠0 ), the equation (1.1) becomes

u t Δ u + 0 t g ( t s ) Δ u ( s ) d s = | u | p 2 u , (1.9)

it arises from the study of heat conduction in materials with memory. In this case, Messaoudi [22,23] obtained a blow-up result for certain solutions with positive or negative initial energy and Giorgi [24] got the asymptotic behavior. Furthermore, for the quasilinear case

| u t | l 2 u t Δ u + 0 t g ( t s ) Δ u ( s ) d s = | u | p 2 u , (1.10)

Messaoudi et al. [25,26] established a general decay result from which the usual exponential and polynomial decay results are just only special cases for (1.10) without reaction term |u|p−2u. Liu et al. [27] obtained a general decay of the energy function for the global solution and a blow-up result for the solution with both positive and negative initial energy for (1.10).

In this paper, we will investigate problem (1.1)–(1.3), which has few result of the problem to our knowledge. We will prove that Lp-norm of the solution grows as an exponential function. An essential tool to the proof is an idea used in [28,29], which based on an auxiliary function (a small perturbation of the total energy), using a differential inequality and obtaining the result. This result extend the early paper. This article is organized as follows. Section 2 is concerned with some notations and statement of assumptions. In Section 3, we give the main result.

2. Preliminaries

In this section, we will give some notations and statement of assumptions for m, p, g. We denote Lp(Ω) by Lp and H01(Ω) by H01,the usual Soblev space. The norm and inner of Lp(Ω) are denoted by ||·||p = ||·||Lp(Ω) and (u, v) = ∫Ωu(x)v(x)dx respectively. Especially, ||·|| = ||·||L2(Ω) for p = 2.

For the relaxation function g and the number m and p, we assume that

(A1) g : R+arrowR+ is a differentiable function satisfying

g ( 0 ) > 0 , 1 0 + g ( s ) d s = l > 0 ;

(A2) there exists a nonincreasing function ξ : R+arrowR+ such that

g ' ( s ) ξ ( s ) g ( s ) ;

(A3) we also assume that

2 < k < p 2 ( n 1 ) n 2 , if n 3 ; 2 < k < p < + , if n = 1 , 2 .

Similar to [15,27], we call u(x, t) a weak solution to problem (1.1)–(1.3) on Ω×[0, T), if

u C ( 0 , T ; H 0 1 ) C 1 ( 0 , T ; L 2 ) , | u | k 2 u t L 2 ( Ω × [ 0 , T ) )

satisfying u(x, 0) = u0(x) and

0 t Ω [ u ( s ) v ( s ) 0 s g ( s τ ) u ( τ ) v ( τ ) d τ + u t ( s ) v ( s ) + | u | k 2 u t v | u | p 2 u v ] d x d s = 0 , v C ( 0 , T ; H 0 1 ) , t [ 0 , T ) .

In this paper, we always assume that problem (1.1)–(1.3) exists a local solution (see [16,17]).

Now, we introduce two functionals

E ( t ) = E ( u ) = 1 2 | | u | | 2 + 1 2 ( g u ) ( t ) 1 2 0 t g ( s ) d s | | u | | 2 1 p | | u | | p p , (2.1)

E ( 0 ) = 1 2 | | u 0 | | 2 1 p | | u 0 | | p p , (2.2)

where uH01 and

( g v ) ( t ) = 0 t g ( t s ) | | v ( s ) v ( t ) | | 2 d s .

Multiplying Equation (1.1) by ut and integrating over Ω, by (A2)(g'(t)≤0), we have

E ' ( t ) = | | u t | | 2 + 1 2 ( g ' u ) ( t ) Ω | u | k 2 u t 2 d x 1 2 g ( t ) | | u | | 2 0 . (2.3)

3. Exponential Growth of Solution

In this section, we prove the main result. Our technique is different from that in [5][15,27][28,29] because of the presence of nonlinear term |u|k−2ut and the memory term.

Theorem

Suppose that assumptions (A1), (A2) and (A3) hold, u0H01 and u is a local solution of the system (1.1)–(1.3), and E(0)<0. Furthermore, we assume that

0 g ( s ) d s < p 2 p 1 . (3.1)

Then the solution of the system (1.1)–(1.3) grows exponentially.

Proof

We set

H ( t ) = E ( t ) . (3.2)

By the definition of H(t) and (2.3)

H ' ( t ) = E ' ( t ) 0 . (3.3)

Consequently, by E(0)<0, we have

H ( 0 ) = E ( 0 ) > 0 . (3.4)

Noting that

H ( t ) 1 p | | u | | p p = [ 1 2 | | u | | 2 + 1 2 ( g u ) ( t ) 1 2 0 t g ( s ) d s | | u | | 2 ] < 0 , (3.5)

then (3.3), (3.4) and (3.5) imply

0 < H ( 0 ) H ( t ) 1 p | | u | | p p . (3.6)

Let us define the function

L ( t ) = H ( t ) + ϵ 2 | | u | | 2 . (3.7)

By taking the time derivative of (3.7) and by (1.1), we have

L ' ( t ) = H ' ( t ) + ϵ Ω u u t d x = | | u t | | 2 1 2 ( g ' u ) ( t ) + Ω | u | k 2 u t 2 d x + 1 2 g ( t ) | | u | | 2 + ϵ | | u | | p p ϵ | | u | | 2 + ϵ Ω 0 t g ( t s ) u ( t ) u ( s ) d s d x ϵ Ω | u | k 2 u u t d x | | u t | | 2 ϵ | | u | | 2 + ϵ | | u | | p p + ϵ Ω 0 t g ( t s ) u ( t ) u ( s ) d s d x + Ω | u | k 2 u t 2 d x ϵ Ω | u | k 2 u u t d x . (3.8)

To estimate the last term in the right-hand side of (3.8), by using the following Young's inequality

a b δ 1 a 2 + δ b 2 ,

we have

Ω | u | k 2 u u t d x = Ω | u | k 2 2 u t | u | k 2 2 u d x δ 1 Ω | u | k 2 u t 2 d x + δ Ω | u | k d x .

Therefore, combining with

Ω 0 t g ( t s ) u ( t ) u ( s ) d s d x = 0 t g ( s ) d s | | u ( t ) | | 2 + 0 t g ( t s ) Ω u ( t ) ( u ( s ) u ( t ) ) d x d s 1 2 0 t g ( s ) d s | | u ( t ) | | 2 1 2 ( g u ) ( t ) ,

we have

L ' ( t ) | | u t | | 2 + ϵ ( 1 2 0 t g ( s ) d s 1 ) | | u | | 2 + ϵ | | u | | p p ϵ δ | | u | | k k + ( 1 ϵ δ 1 ) Ω | u | k 2 u t 2 d x ϵ 2 ( g u ) . (3.9)

By using

| | u | | p p = p H ( t ) + p 2 ( 1 0 t g ( s ) d s ) | | u | | 2 + p 2 ( g u ) ,

(3.9) becomes

L ' ( t ) | | u t | | 2 + ϵ ( 1 2 0 t g ( s ) d s 1 ) | | u | | 2 + ϵ [ p H ( t ) + p 2 ( g u ) + p 2 ( 1 0 t g ( s ) d s ) | | u | | 2 ] ϵ δ | | u | | k k + ( 1 ϵ δ 1 ) Ω | u | k 2 u t 2 d x ϵ 2 ( g u ) | | u t | | 2 + ( 1 ϵ δ 1 ) Ω | u | k 2 u t 2 d x + ϵ a 1 | | u | | 2 + ϵ a 2 ( g u ) + ϵ p H ( t ) ϵ δ | | u | | k k , (3.10)

where a 1 = 1 - p 2 0 g ( s ) d s + p - 2 2 > 0 by (3.1), a 2 = p 2 - 1 > 0 . Noting that p>k>2 and embedding theorem, we have

| | u | | k k C | | u | | p k C ( | | u | | p p ) k p , (3.11)

where C>0 is a positive embedding constant. Since 0 < k p < 1 , now applying the inequality x l ( x + 1 ) ( 1 + 1 z ) ( x + z ) , which holds for all x≥0, 0≤l≤1, z>0, in particular, taking x = | | u | | p p , l = k p , z = H ( 0 ) , we obtain

( | | u | | p p ) k p ( 1 + 1 H ( 0 ) ) ( | | u | | p p + H ( 0 ) ) ,

then from (3.6) and (3.11)

| | u | | k k C | | u | | p k C 1 | | u | | p p ,

we have

L ' ( t ) | | u t | | 2 + ( 1 ϵ δ 1 ) Ω | u | k 2 u t 2 d x + ϵ a 1 | | u | | 2 + ϵ a 2 ( g u ) + ϵ p H ( t ) ϵ δ C 1 | | u | | p p , (3.12)

Taking 0<a3<min{a1, a2}, and by 2 H ( t ) - | | u | | 2 - ( g u ) + 2 p | | u | | p p , we have

L ' ( t ) | | u t | | 2 + ( 1 ϵ δ 1 ) Ω | u | k 2 u t 2 d x + ϵ ( a 1 a 3 ) | | u | | 2 + ϵ p H ( t ) + ϵ ( a 2 a 3 ) ( g u ) + ϵ ( 2 p a 2 δ C 1 ) | | u | | p p + ϵ a 3 [ | | u | | 2 + ( g u ) 2 p | | u | | p p ] = | | u t | | 2 + ( 1 ϵ δ 1 ) Ω | u | k 2 u t 2 d x + ϵ ( a 1 a 3 ) | | u | | 2 + ϵ ( a 2 a 3 ) ( g u ) + ϵ ( 2 p a 3 δ C 1 ) | | u | | p p + ϵ ( p 2 a 3 ) H ( t ) . (3.13)

Taking δ small enough such that 2 p a 3 - δ C 1 > 0 , and then taking ϵ small enough such that 1 − ϵδ−1>0, and noting that p − 2a3>0, then

L ' ( t ) C 2 ( H ( t ) + | | u t | | 2 + | | u | | 2 + ( g u ) + | | u | | p p ) . (3.14)

On the other hand, by the definition of H(t), we get

L ' ( t ) H ( t ) + | | u | | 2 C 3 ( H ( t ) + | | u | | 2 ) C 3 ( H ( t ) + | | u t | | 2 + | | u | | 2 + ( g u ) + | | u | | p p ) . (3.15)

From (3.14) and (3.15), we obtain the differential inequality

L ' ( t ) r L ( t ) . (3.16)

Integration of (3.16) between 0 and t gives us

L ( t ) L ( 0 ) exp ( r t ) . (3.17)

From (3.8) and with ϵ small enough, we have

L ( t ) H ( t ) 1 p | | u | | p p . (3.18)

By (3.17) and (3.18), we deduce

| | u | | p p C exp ( r t ) .

Therefore, we conclude that the solution in the Lp-norm growths exponentially.

Remark

By the same method (similar to [29]), we can also get the similar result to problem (1.1)–(1.3) with positive initial energy.

Concluding

In this paper, the initial boundary value problem of a class of reaction diffusion equation with memory and multiple nonlinearities is considered. Using a differential inequality, exponential growth of solution with Lp-norm is proved for negative and positive initial energy.

Competing Interests

The authors declare no competing interests.

Acknowledgement

This work is supported by National Natural Science Foundation of China (No.11601122).

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Research Article
Research in Applied Mathematics
Vol. 1 (2017), Article ID 101258, 9 pages
doi:10.11131/2017/101258

Exponential Growth of Solution for a Class of Reaction Diffusion Equation with Memory and Multiple Nonlinearities

Jian Dang, Qingying Hu, Suxia Xia, and Hongwei Zhang

Department of Mathematics, Henan University of Technology, Zhengzhou 450001, China

Received 8 November 2016; Accepted 19 August 2017

Editor: Zhouhong Li

Copyright © 2017 Jian Dang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the initial boundary value problem of a class of reaction diffusion equation with memory and multiple nonlinearities. We show the exponential growth of solution with Lp-norm using a differential inequality.