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

#### 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 *L*_{p}-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

${u}_{t}-\Delta u+{\int}_{0}^{t}g(t-s)\Delta u\left(s\right)ds+{\left|u\right|}^{k-2}{u}_{t}={\left|u\right|}^{p-2}u,$ | (1.1) |

$u(x,t)=0,x\in \partial \Omega ,$ | (1.2) |

$u(x,0)={u}_{0}\left(x\right),x\in \Omega ,$ | (1.3) |

where *k*>2, *p*>2 are real numbers and *Ω* is bounded domain in *R*^{n}
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−2}*u*_{t}, and
when the function *g* vanishes identically (i.e. *g* = 0), then the
equation (1.1) can be reduced to the following equation

${u}_{t}-\Delta u={\left|u\right|}^{p-2}u,$ | (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−2}*u*
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

${u}_{t}{-\mathrm{div}\hspace{0.17em}\left(\right|\u25bdu|}^{m-2}\u25bdu)=f\left(u\right),$ | (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

$|{u}_{t}{|}^{l-2}{u}_{t}{-\mathrm{div}\hspace{0.17em}\left(\right|\u25bdu|}^{m-2}\u25bdu)=f\left(u\right),$ | (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]

$\beta {\left(u\right)}_{t}-\Delta u={\left|u\right|}^{p-2}u$ | (1.7) |

if we take *β*(*u*) = *u* + |*u*|^{m−2}*u*. 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

${u}_{t}-{u}_{xx}+{\left|u\right|}^{m-2}{u}_{t}={\left|u\right|}^{p-2}u.$ |

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

${\left({u}^{m}\right)}_{t}-\Delta u=f\left(u\right),$ |

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

${u}_{t}-\Delta u-\Delta {u}_{t}+{\left|u\right|}^{m-2}{u}_{t}=u(u-\alpha )(u-\beta ),$ | (1.8) |

with initial boundary value (1.2) and (1.3) in *R*^{3} 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−2}*u*_{t}, and
with the presence of the viscoelastic term (i.e. *g*≠0 ), the equation (1.1) becomes

${u}_{t}-\Delta u+{\int}_{0}^{t}g(t-s)\Delta u\left(s\right)ds={\left|u\right|}^{p-2}u,$ | (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

$|{u}_{t}{|}^{l-2}{u}_{t}-\Delta u+{\int}_{0}^{t}g(t-s)\Delta u\left(s\right)ds={\left|u\right|}^{p-2}u,$ | (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−2}*u*. 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 *L*_{p}-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 *L*^{p}(*Ω*) by
*L*^{p} and *H*_{0}^{1}(*Ω*) by *H*_{0}^{1},the usual Soblev space. The norm and inner of
*L*^{p}(*Ω*) are denoted by
||·||_{p} = ||·||_{Lp(Ω)} and
(*u*, *v*) = ∫_{Ω}*u*(*x*)*v*(*x*)*d**x* respectively. Especially,
||·|| = ||·||_{L2(Ω)} for *p* = 2.

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

(A1)
*g* : *R*^{+}*a**r**r**o**w**R*^{+} is a differentiable function satisfying

$g\left(0\right)>0,1-{\int}_{0}^{+\infty}g\left(s\right)ds=l>0;$ |

(A2) there exists a nonincreasing function *ξ* : *R*^{+}*a**r**r**o**w**R*^{+}
such that

${g}^{\text{'}}\left(s\right)\le -\xi \left(s\right)g\left(s\right);$ |

(A3) we also assume that

$2<k<p\le \frac{2(n-1)}{n-2},\mathrm{if}n\ge 3;2<k<p<+\infty ,\mathrm{if}n=1,2.$ |

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

$u\in C(0,T;{H}_{0}^{1})\cap {C}^{1}(0,T;{L}^{2}),{\left|u\right|}^{k-2}{u}_{t}\in {L}^{2}(\Omega \times [0,T))$ |

satisfying *u*(*x*, 0) = *u*_{0}(*x*) and

$\begin{array}{cc}& {\displaystyle {\int}_{0}^{t}{\int}_{\Omega}[\u25bdu\left(s\right)\u25bdv\left(s\right)-{\int}_{0}^{s}g(s-\tau )\u25bdu\left(\tau \right)\u25bdv\left(\tau \right)d\tau +{u}_{t}\left(s\right)v\left(s\right)}\hfill \\ & {\displaystyle +{\left|u\right|}^{k-2}{u}_{t}v-{\left|u\right|}^{p-2}uv]dxds=0,\forall v\in C(0,T;{H}_{0}^{1}),\forall t\in [0,T).}\hfill \end{array}$ |

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

Now, we introduce two functionals

$E\left(t\right)=E\left(u\right)=\frac{1}{2}\left|\right|\u25bd{u\left|\right|}^{2}+\frac{1}{2}(g\otimes \u25bdu)\left(t\right)-\frac{1}{2}{\int}_{0}^{t}g\left(s\right)ds\left|\right|\u25bd{u\left|\right|}^{2}-\frac{1}{p}{\left|\right|u\left|\right|}_{p}^{p},$ | (2.1) |

$E\left(0\right)=\frac{1}{2}\left|\right|\u25bd{u}_{0}{\left|\right|}^{2}-\frac{1}{p}\left|\right|{u}_{0}{\left|\right|}_{p}^{p},$ | (2.2) |

where *u*∈*H*_{0}^{1} and

$(g\otimes \u25bdv)\left(t\right)={\int}_{0}^{t}g(t-s)\left|\right|\u25bdv\left(s\right)-\u25bd{v\left(t\right)\left|\right|}^{2}ds.$ |

Multiplying Equation (1.1) by *u*_{t} and integrating over *Ω*, by (A2)(*g*^{'}(*t*)≤0), we have

${E}^{\text{'}}\left(t\right)=-\left|\right|{u}_{t}{\left|\right|}^{2}+\frac{1}{2}({g}^{\text{'}}\otimes \u25bdu)\left(t\right)-{\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx-\frac{1}{2}g\left(t\right)\left|\right|\u25bd{u\left|\right|}^{2}\le 0.$ | (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−2}*u*_{t} and the memory term.

Theorem

Suppose that assumptions (A1), (A2) and (A3) hold, *u*_{0}∈*H*_{0}^{1}
and *u* is a local solution of the system (1.1)–(1.3), and *E*(0)<0. Furthermore, we assume that

${\int}_{0}^{\infty}g\left(s\right)ds<\frac{p-2}{p-1}.$ | (3.1) |

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

Proof

We set

$H\left(t\right)=-E\left(t\right).$ | (3.2) |

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

${H}^{\text{'}}\left(t\right)=-{E}^{\text{'}}\left(t\right)\ge 0.$ | (3.3) |

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

$H\left(0\right)=-E\left(0\right)>0.$ | (3.4) |

Noting that

$H\left(t\right)-\frac{1}{p}{\left|\right|u\left|\right|}_{p}^{p}=-[\frac{1}{2}\left|\right|\u25bd{u\left|\right|}^{2}+\frac{1}{2}(g\otimes \u25bdu)\left(t\right)-\frac{1}{2}{\int}_{0}^{t}g\left(s\right)ds\left|\right|\u25bd{u\left|\right|}^{2}]<0,$ | (3.5) |

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

$0<H\left(0\right)\le H\left(t\right)\le \frac{1}{p}{\left|\right|u\left|\right|}_{p}^{p}.$ | (3.6) |

Let us define the function

$L\left(t\right)=H\left(t\right)+\frac{\u03f5}{2}{\left|\right|u\left|\right|}^{2}.$ | (3.7) |

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

$\begin{array}{cc}{L}^{\text{'}}\left(t\right)\hfill & {\displaystyle ={H}^{\text{'}}\left(t\right)+\u03f5{\int}_{\Omega}u{u}_{t}dx}\hfill \\ & {\displaystyle =\left|\right|{u}_{t}{\left|\right|}^{2}-\frac{1}{2}({g}^{\text{'}}\otimes \u25bdu)\left(t\right)+{\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx+\frac{1}{2}g\left(t\right)\left|\right|\u25bd{u\left|\right|}^{2}}\hfill \\ & {\displaystyle +{\u03f5\left|\right|u\left|\right|}_{p}^{p}-\u03f5\left|\right|\u25bd{u\left|\right|}^{2}+\u03f5{\int}_{\Omega}{\int}_{0}^{t}g(t-s)\u25bdu\left(t\right)\u25bdu\left(s\right)dsdx}\hfill \\ & {\displaystyle -\u03f5{\int}_{\Omega}{\left|u\right|}^{k-2}u{u}_{t}dx}\hfill \\ & {\displaystyle \ge \left|\right|{u}_{t}{\left|\right|}^{2}-\u03f5\left|\right|\u25bd{u\left|\right|}^{2}+{\u03f5\left|\right|u\left|\right|}_{p}^{p}+\u03f5{\int}_{\Omega}{\int}_{0}^{t}g(t-s)\u25bdu\left(t\right)\u25bdu\left(s\right)dsdx}\hfill \\ & {\displaystyle +{\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx-\u03f5{\int}_{\Omega}{\left|u\right|}^{k-2}u{u}_{t}dx.}\hfill \end{array}$ | (3.8) |

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

$ab\le {\delta}^{-1}{a}^{2}+\delta {b}^{2},$ |

we have

$\begin{array}{cc}{\displaystyle {\int}_{\Omega}{\left|u\right|}^{k-2}u{u}_{t}dx}\hfill & {\displaystyle ={\int}_{\Omega}{\left|u\right|}^{\frac{k-2}{2}}{u}_{t}{\left|u\right|}^{\frac{k-2}{2}}udx}\hfill \\ & {\displaystyle \le {\delta}^{-1}{\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx+\delta {\int}_{\Omega}{\left|u\right|}^{k}dx.}\hfill \end{array}$ |

Therefore, combining with

$\begin{array}{cc}& {\displaystyle {\int}_{\Omega}{\int}_{0}^{t}g(t-s)\u25bdu\left(t\right)\u25bdu\left(s\right)dsdx}\hfill \\ & {\displaystyle ={\int}_{0}^{t}g\left(s\right)ds\left|\right|\u25bd{u\left(t\right)\left|\right|}^{2}+{\int}_{0}^{t}g(t-s){\int}_{\Omega}\u25bdu\left(t\right)(\u25bdu\left(s\right)-\u25bdu\left(t\right))dxds}\hfill \\ & {\displaystyle \ge \frac{1}{2}{\int}_{0}^{t}g\left(s\right)ds\left|\right|\u25bd{u\left(t\right)\left|\right|}^{2}-\frac{1}{2}(g\otimes \u25bdu)\left(t\right),}\hfill \end{array}$ |

we have

$\begin{array}{cc}{\displaystyle {L}^{\text{'}}\left(t\right)}\hfill & {\displaystyle \ge \left|\right|{u}_{t}{\left|\right|}^{2}+\u03f5(\frac{1}{2}{\int}_{0}^{t}g\left(s\right)ds-1)\left|\right|\u25bd{u\left|\right|}^{2}+{\u03f5\left|\right|u\left|\right|}_{p}^{p}}\hfill \\ & {\displaystyle -{\u03f5\delta \left|\right|u\left|\right|}_{k}^{k}+(1-\u03f5{\delta}^{-1}){\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx-\frac{\u03f5}{2}(g\otimes \u25bdu).}\hfill \end{array}$ | (3.9) |

By using

${\left|\right|u\left|\right|}_{p}^{p}=pH\left(t\right)+\frac{p}{2}(1-{\int}_{0}^{t}g\left(s\right)ds)\left|\right|\u25bd{u\left|\right|}^{2}+\frac{p}{2}(g\otimes \u25bdu),$ |

(3.9) becomes

$\begin{array}{cc}{L}^{\text{'}}\left(t\right)\hfill & {\displaystyle \ge \left|\right|{u}_{t}{\left|\right|}^{2}+\u03f5(\frac{1}{2}{\int}_{0}^{t}g\left(s\right)ds-1)\left|\right|\u25bd{u\left|\right|}^{2}}\hfill \\ & {\displaystyle +\u03f5[pH\left(t\right)+\frac{p}{2}(g\otimes \u25bdu)+\frac{p}{2}(1-{\int}_{0}^{t}g\left(s\right)ds)||\u25bdu|{|}^{2}]}\hfill \\ & {\displaystyle -{\u03f5\delta \left|\right|u\left|\right|}_{k}^{k}+(1-\u03f5{\delta}^{-1}){\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx-\frac{\u03f5}{2}(g\otimes \u25bdu)}\hfill \\ & {\displaystyle \ge \left|\right|{u}_{t}{\left|\right|}^{2}+(1-\u03f5{\delta}^{-1}){\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx}\hfill \\ & {\displaystyle +\u03f5{a}_{1}\left|\right|\u25bd{u\left|\right|}^{2}+\u03f5{a}_{2}(g\otimes \u25bdu)+\u03f5pH\left(t\right)-{\u03f5\delta \left|\right|u\left|\right|}_{k}^{k},}\hfill \end{array}$ | (3.10) |

where
${a}_{1}=\frac{1-p}{2}{\int}_{0}^{\infty}g\left(s\right)ds+\frac{p-2}{2}>0$ by (3.1),
${a}_{2}=\frac{p}{2}-1>0$.
Noting that *p*>*k*>2 and embedding theorem, we have

${\left|\right|u\left|\right|}_{k}^{k}\le {C\left|\right|u\left|\right|}_{p}^{k}\le {C\left(\right|\left|u\right||}_{p}^{p}{)}^{\frac{k}{p}},$ | (3.11) |

where *C*>0 is a positive embedding constant.
Since $0<\frac{k}{p}<1$, now applying the inequality
${x}^{l}\le (x+1)\le (1+\frac{1}{z})(x+z),$ which holds for all *x*≥0, 0≤*l*≤1, *z*>0,
in particular, taking
$x={\left|\right|u\left|\right|}_{p}^{p},l=\frac{k}{p},z=H\left(0\right)$, we obtain

${\left(\right|\left|u\right||}_{p}^{p}{)}^{\frac{k}{p}}\le (1+\frac{1}{H\left(0\right)}){\left(\right|\left|u\right||}_{p}^{p}+H\left(0\right)),$ |

then from (3.6) and (3.11)

${\left|\right|u\left|\right|}_{k}^{k}\le {C\left|\right|u\left|\right|}_{p}^{k}\le {C}_{1}{\left|\right|u\left|\right|}_{p}^{p},$ |

we have

$\begin{array}{cc}{L}^{\text{'}}\left(t\right)\hfill & {\displaystyle \ge \left|\right|{u}_{t}{\left|\right|}^{2}+(1-\u03f5{\delta}^{-1}){\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx}\hfill \\ & {\displaystyle +\u03f5{a}_{1}\left|\right|\u25bd{u\left|\right|}^{2}+\u03f5{a}_{2}(g\otimes \u25bdu)+\u03f5pH\left(t\right)-\u03f5\delta {C}_{1}{\left|\right|u\left|\right|}_{p}^{p},}\hfill \end{array}$ | (3.12) |

Taking 0<*a*_{3}<min{*a*_{1}, *a*_{2}}, and by
$2H\left(t\right)\ge -\left|\right|\u25bd{u\left|\right|}^{2}-(g\otimes \u25bdu)+\frac{2}{p}{\left|\right|u\left|\right|}_{p}^{p}$, we have

$\begin{array}{cc}{L}^{\text{'}}\left(t\right)\hfill & {\displaystyle \ge \left|\right|{u}_{t}{\left|\right|}^{2}+(1-\u03f5{\delta}^{-1}){\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx+\u03f5({a}_{1}-{a}_{3})\left|\right|\u25bd{u\left|\right|}^{2}+\u03f5pH\left(t\right)}\hfill \\ & {\displaystyle +\u03f5({a}_{2}-{a}_{3})(g\otimes \u25bdu)+\u03f5(\frac{2}{p}{a}_{2}-\delta {C}_{1}){\left|\right|u\left|\right|}_{p}^{p}}\hfill \\ & {\displaystyle +\u03f5{a}_{3}\left[\right||\u25bd{u\left|\right|}^{2}+(g\otimes \u25bdu)-\frac{2}{p}{\left|\right|u\left|\right|}_{p}^{p}]}\hfill \\ & {\displaystyle =\left|\right|{u}_{t}{\left|\right|}^{2}+(1-\u03f5{\delta}^{-1}){\int}_{\Omega}{\left|u\right|}^{k-2}{u}_{t}^{2}dx+\u03f5({a}_{1}-{a}_{3})\left|\right|\u25bd{u\left|\right|}^{2}}\hfill \\ & {\displaystyle +\u03f5({a}_{2}-{a}_{3})(g\otimes \u25bdu)+\u03f5(\frac{2}{p}{a}_{3}-\delta {C}_{1}){\left|\right|u\left|\right|}_{p}^{p}+\u03f5(p-2{a}_{3})H\left(t\right).}\hfill \end{array}$ | (3.13) |

Taking *δ* small enough such that $\frac{2}{p}{a}_{3}-\delta {C}_{1}>0$,
and then taking *ϵ* small enough such that 1 − *ϵ**δ*^{−1}>0,
and noting that *p* − 2*a*_{3}>0, then

${L}^{\text{'}}\left(t\right)\ge {C}_{2}(H\left(t\right)+||{u}_{t}{\left|\right|}^{2}+\left|\right|\u25bd{u\left|\right|}^{2}+(g\otimes \u25bdu)+{\left|\right|u\left|\right|}_{p}^{p}).$ | (3.14) |

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

$\begin{array}{cc}{L}^{\text{'}}\left(t\right)\hfill & {\displaystyle \le H\left(t\right)+{\left|\right|u\left|\right|}^{2}\le {C}_{3}{(H\left(t\right)+||\u25bdu||}^{2})}\hfill \\ & {\displaystyle \le {C}_{3}(H\left(t\right)+||{u}_{t}{\left|\right|}^{2}+\left|\right|\u25bd{u\left|\right|}^{2}+(g\otimes \u25bdu)+{\left|\right|u\left|\right|}_{p}^{p}).}\hfill \end{array}$ | (3.15) |

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

${L}^{\text{'}}\left(t\right)\ge rL\left(t\right).$ | (3.16) |

Integration of (3.16) between 0 and t gives us

$L\left(t\right)\ge L\left(0\right)\mathrm{exp}\hspace{0.17em}\left(rt\right).$ | (3.17) |

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

$L\left(t\right)\le H\left(t\right)\le \frac{1}{p}{\left|\right|u\left|\right|}_{p}^{p}.$ | (3.18) |

By (3.17) and (3.18), we deduce

${\left|\right|u\left|\right|}_{p}^{p}\ge C\mathrm{exp}\hspace{0.17em}\left(rt\right).$ |

Therefore, we conclude that the solution in the *L*_{p}-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 *L*_{p}-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).

#### References

- Z. Jiang, S. Zheng, and X. Song, “Blow-up analysis for a nonlinear diffusion equation with nonlinear boundary conditions,”
, vol. 17, no. 2, pp. 193–199, 2004. Publisher Full Text | Google Scholar*Applied Mathematics Letters. An International Journal of Rapid Publication* - K. Deng and H. A. Levine, “The role of critical exponents in blow-up theorems: the sequel,”
, vol. 243, no. 1, pp. 85–126, 2000. Publisher Full Text | Google Scholar*Journal of Mathematical Analysis and Applications* - H. A. Levine, “Some nonexistence and instability theorems for solutions of formally parabolic equations of the form
*Pu*t = −*Au*+*F*(*u*),”, vol. 51, pp. 371–386, 1973. Publisher Full Text | Google Scholar*Archive for Rational Mechanics and Analysis* - V. K. Kalantarov and O. A. Ladyzhenskaya, “The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types,”
, vol. 10, no. 1, pp. 53–70, 1978. Publisher Full Text | Google Scholar*Journal of Soviet Mathematics* - H. A. Levine, S. R. Park, and J. Serrin, “Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type,”
, vol. 142, no. 1, pp. 212–229, 1998. Publisher Full Text | Google Scholar*Journal of Differential Equations* - S. A. Messaoudi, “A note on blow up of solutions of a quasilinear heat equation with vanishing initial energy,”
, vol. 273, no. 1, pp. 243–247, 2002. Publisher Full Text | Google Scholar*Journal of Mathematical Analysis and Applications* - W. Liu and M. Wang, “Blow-up of the solution for a
*p*-Laplacian equation with positive initial energy,”, vol. 103, no. 2, pp. 141–146, 2008. Publisher Full Text | Google Scholar*Acta Applicandae Mathematicae* - C. Ayouch, E. H. Essoufi, and M. Tilioua, “A finite difference scheme for the timefractional Landau-Lifshitz-Bloch equation,”
, vol. 1, pp. 1–10, 2017. Publisher Full Text | Google Scholar*Research in Applied Mathematics* - P. Pucci and J. Serrin, “Asymptotic stability for nonlinear parabolic systems,” in
, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1996.*Energy Methods in Continuum Mechanics* - J.-S. Pang and H.-W. Zhang, “Existence and nonexistence of the global solution on the quasilinear parabolic equation,”
, vol. 22, no. 3, pp. 444–450, 2007.*Chinese Quarterly Journal of Mathematics* - J. S. Pang and Q. Y. Hu, “Global nonexistence for a class of quasilinear parabolic equation with source term and positive initial energy,”
, vol. 37, no. 5, pp. 448–451, 2007 (Chinese).*Journal of Henan University (Natural Science)* - S. Berrimi and S. A. Messaoudi, “A decay result for a quasilinear parabolic system,” in
, vol. 63 of*Elliptic and Parabolic Problems**Progress in Nonlinear Differential Equations and Their Applications*, pp. 43–50, 2005.Zbl1082.35029 - A. Eden, B. Michaux, and J.-M. Rakotoson, “Doubly nonlinear parabolic-type equations as dynamical systems,”
, vol. 3, no. 1, pp. 87–131, 1991. Publisher Full Text | Google Scholar*Journal of Dynamics and Differential Equations* - H. El Ouardi and A. El Hachimi, “Attractors for a class of doubly nonlinear parabolic systems,”
, pp. 1–15, 2006. Publisher Full Text | Google Scholar*Electronic Journal of Qualitative Theory of Differential Equations* - N. Polat, “Blow up of solution for a nonlinear reaction diffusion equation with multiple nonlinearities,” vol. 2, pp. 123–128, 2007.
- M. O. Korpusov and A. G. Sveshnikov, “Sufficient conditions, which are close to necessary conditions, for the blow-up of the solution of a strongly nonlinear generalized Boussinesq equation,”
, vol. 48, no. 9, pp. 1629–1637, 2008. Publisher Full Text | Google Scholar*Computational Mathematics and Mathematical Physics* - A. B. Al'shin, M. O. Korpusov, and A. G. Sveshnikov,
, De Gruyter, 2011.*Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations* - H. Zhang, J. Lu, and Q. Hu, “Exponential growth of solution of a strongly nonlinear generalized Boussinesq equation,”
, vol. 68, no. 12, part A, pp. 1787–1793, 2014. Publisher Full Text | Google Scholar*Computers \& Mathematics with Applications. An International Journal* - L. X. Truong and N. Van Y, “Exponential growth with
*L**p*-norm of solutions for nonlinear heat equations with viscoelastic term,”, vol. 273, pp. 656–663, 2016. Publisher Full Text | Google Scholar*Applied Mathematics and Computation* - L. X. Truong and N. V. Y, “On a class of nonlinear heat equations with viscoelastic term,”
, vol. 72, no. 1, pp. 216–232, 2016. Publisher Full Text | Google Scholar*Computers \& Mathematics with Applications. An International Journal* - H. W. Zhang, G. X. Zhang, and Q. Y. Hu, “Global nonexistence of solutions for a class of doubly nonlinear parabolic equations,”
, vol. 18, no. 2, pp. 204–211, 2016.*Acta Analysis Functionalis Applicata. AAFA. Yingyong Fanhanfenxi Xuebao* - S. A. Messaoudi, “Blow-up of solutions of a semi linear heat equation with a Visco-elastic term,” in
, H. Brezis, M. Chipot, and J. Escher, Eds., vol. 64 of*Nonlinear Elliptic and Parabolic Problems**Progress in Nonlinear Differential Equations and Their Applications*, pp. 351–356, Birkhäuser, Basel, Switzerland, 2005. - S. A. Messaoudi, “Blow-up of solutions of a semilinear heat equation with a memory term,”
, no. 2, pp. 87–94, 2005. Publisher Full Text | Google Scholar*Abstract and Applied Analysis* - C. Giorgi, V. Pata, and A. Marzocchi, “Asymptotic behavior of a semilinear problem in heat conduction with memory,”
, vol. 5, no. 3, pp. 333–354, 1998. Publisher Full Text | Google Scholar*NoDEA. Nonlinear Differential Equations and Applications* - S. A. Messaoudi and B. Tellab, “A general decay result in a quasilinear parabolic system with viscoelastic term,”
, vol. 25, no. 3, pp. 443–447, 2012. Publisher Full Text | Google Scholar*Applied Mathematics Letters. An International Journal of Rapid Publication* - S. A. Messaoudi, “General decay of the solution energy in a viscoelastic equation with a nonlinear source,”
, vol. 69, no. 8, pp. 2589–2598, 2008. Publisher Full Text | Google Scholar*Nonlinear Analysis. Theory, Methods \& Applications. An International Multidisciplinary Journal* - G. Liu and H. Chen, “Global and blow-up of solutions for a quasilinear parabolic system with viscoelastic and source terms,”
, vol. 37, no. 1, pp. 148–156, 2014. Publisher Full Text | Google Scholar*Mathematical Methods in the Applied Sciences* - S. Gerbi and B. Said-Houari, “Local existence and exponential growth for a semilinear damped wave equation with dynamic boundary conditions,”
, vol. 13, no. 11-12, pp. 1051–1074, 2008.*Advances in Differential Equations* - B. Said-Houari, “Exponential growth of positive initial-energy solutions of a system of nonlinear viscoelastic wave equations with damping and source terms,”
, vol. 62, no. 1, pp. 115–133, 2011. Publisher Full Text | Google Scholar*Z. Angew. Math. Phys*

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

#### 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 *L*_{p}-norm using a differential inequality.