# Energy decay and nonexistence of solution for a reaction-diffusion equation with exponential nonlinearity

- Huiya Dai
^{1, 2}Email author and - Hongwei Zhang
^{2}

**2014**:70

https://doi.org/10.1186/1687-2770-2014-70

© Dai and Zhang; licensee Springer. 2014

**Received: **2 December 2013

**Accepted: **13 March 2014

**Published: **25 March 2014

## Abstract

In this work we consider the energy decay result and nonexistence of global solution for a reaction-diffusion equation with generalized Lewis function and nonlinear exponential growth. There are very few works on the reaction-diffusion equation with exponential growth *f* as a reaction term by potential well theory. The ingredients used are essentially the Trudinger-Moser inequality.

## Keywords

## 1 Introduction

here $f(s)$ is a reaction term with exponential growth at infinity to be specified later, Ω is a bounded domain with smooth boundary *∂* Ω in ${R}^{2}$.

For the reaction-diffusion equation with polynomial growth reaction terms (that is, equation (1) with $a(x,t)=1$ and $f(u)={|u|}^{p-1}u$), there have been many works in the literature; one can find a review of previous results in [1, 2] and references therein, which are not listed in this paper just for concision. Problem (1)-(3) with $a(x,t)>0$ describes the chemical reaction processes accompanied by diffusion [2]. The author of work [1] proved the existence and asymptotic estimates of global solutions and finite time blow-up of problem (1)-(3) with $a(x,t)>0$ and the critical Sobolev exponent $p=\frac{n+2}{n-2}$ for $f(u)={u}^{p}$.

In this paper we assume that $f(s)$ is a reaction term with exponential growth like ${e}^{{s}^{2}}$ at infinity. When $a(x,t)=1$, $f(u)={e}^{u}$, model (1)-(3) was proposed by [3] and [4]. In this case, Fujita [5] studied the asymptotic stability of the solution. Peral and Vazquez [6] and Pulkkinen [7] considered the stability and blow-up of the solution. Tello [8] and Ioku [9] considered the Cauchy problem of heat equation with $f(u)\approx {e}^{{u}^{2}}$ for $|u|\ge 1$.

Recently, Alves and Cavalcanti [10] were concerned with the nonlinear damped wave equation with exponential source. They proved global existence as well as blow-up of solutions in finite time by taking the initial data inside the potential well [11]. Moreover, they also got the optimal and uniform decay rates of the energy for global solutions.

Motivated by the ideas of [1, 10], we concentrate on studying the uniform decay estimate of the energy and finite time blow-up property of problem (1)-(3) with generalized Lewis function $a(x,t)$ and exponential growth *f* as a reaction term. To the authors’ best knowledge, there are very few works in the literature that take into account the reaction-diffusion equation with exponential growth *f* as a reaction term by potential well theory. The majority of works in the literature make use of the potential well theory when *f* possesses polynomial growth. See, for instance, the works [12–16] and a long list of references therein. The ingredients used in our proof are essentially the Trudinger-Moser inequality (see [17, 18]). We establish decay rates of the energy by considering ideas from the work of Messaoudi [15]. The case of nonexistence results is also treated, where a finite time blow-up phenomenon is exhibited for finite energy solutions by the standard concavity method adapted for our context.

The remainder of our paper is organized as follows. In Section 2 we present the main assumptions and results, Section 3 and Section 4 are devoted to the proof of the main results.

Throughout this study, we denote by $\parallel \cdot \parallel $, ${\parallel \cdot \parallel}_{p}$, ${\parallel \cdot \parallel}_{{H}_{0}^{1}}$ the usual norms in spaces ${L}^{2}(\mathrm{\Omega})$, ${L}^{p}(\mathrm{\Omega})$ and ${H}_{0}^{1}(\mathrm{\Omega})$, respectively.

## 2 Assumptions and preliminaries

In this section, we present the main assumptions and results. We always assume that:

(A1) $a(x,t)$ is a positive differentiable function and is bounded for $t\in [0,+\mathrm{\infty})$, $x\in \mathrm{\Omega}$.

where $F(t)={\int}_{0}^{t}f(s)\phantom{\rule{0.2em}{0ex}}ds$.

A typical example of functions satisfies (A2)-(A4) is $f(t)=C{|t|}^{p-1}t{e}^{M{t}^{\alpha}}$, with given $p>1$, $M>0$, $C>0$, and $\alpha \in (1,2)$.

We also need the following lemmas.

*Let*Ω

*be a bounded domain in*${R}^{2}$.

*For all*$u\in {H}_{0}^{1}(\mathrm{\Omega})$,

*and there exist positive constants*${m}_{2}$

*such that*

**Lemma 2.2** [19]

*Let*$\varphi (t)$

*be a nonincreasing and nonnegative function on*$[0,\mathrm{\infty})$,

*such that*

*then*

*where* *C*, *ω* *are positive constants depending on* $\varphi (0)$ *and other known qualities*.

**Lemma 2.3** [20]

*Suppose that a positive*,

*twice*-

*differentiable function*$H(t)$

*satisfies on*$t\ge 0$

*the inequality*

*where* $\delta >0$, *then there is* ${t}_{1}<{t}_{2}=\frac{H(0)}{\delta {H}^{\prime}(0)}$ *such that* $H(t)\to \mathrm{\infty}$ *as* $t\to {t}_{1}$.

*p*and Ω only such that

Our main results read as follows.

**Theorem 2.1**

*Let*(A1)-(A4)

*hold*.

*Assume further that*${u}_{0}\in {W}_{1}$

*satisfies*

*for some sufficiently small*${\epsilon}_{0}>0$

*and*${C}_{{\epsilon}_{0}}>0$.

*Then there exist positive constants*

*K*

*and*

*k*

*such that the energy*$E(t)$

*satisfies the decay estimates for large*

*t*

**Theorem 2.2** *Let* (A1)-(A4) *hold*. *Assume further that* ${a}_{t}(x,t)\le 0$, ${u}_{0}\in {W}_{2}$ *and* $E(0)<\frac{(\theta -2)d}{\theta}<d$, *then the solutions of* (1)-(3) *blow up in finite time*.

## 3 Proof of decay of the energy

In this section we prove Theorem 2.1. We divide the proof into two lemmas.

**Lemma 3.1** *Under the assumptions of Theorem * 2.1, *we have*, *for all* $t\ge 0$, $u(t)\in {W}_{1}$.

*Proof*Since $I({u}_{0})\ge 0$, then there exists (by continuity) ${T}_{m}<T$ such that

*β*such that $\frac{\theta \beta d}{\theta -2}<\pi $, then, from Trudinger-Moser inequality (11),

This is extended to *T*. □

**Lemma 3.2**

*Under the assumptions of Theorem*2.1,

*we have*,

*for*$\eta =1-[{\epsilon}_{0}{C}_{0}^{2}+{C}_{{\epsilon}_{0}}{C}_{0}^{p}{m}_{2}^{\frac{1}{2}}{(\frac{2\theta}{\theta -2}E(0))}^{p-2}]$,

*Proof*It suffices to rewrite (21) as

Thus (22) follows from (23). □

*Proof of Theorem 2.1*We integrate (15) over $[t,t+1]$ to obtain

*u*and integrate over $\mathrm{\Omega}\times [t,t+1]$ to arrive at

*A*,

*θ*,

*η*only. We then use Young’s inequality to get from (30) and (24)

By (12) in Lemma 2.2 we then get the results. □

## 4 Proof of the blow-up result

In this section, we shall prove Theorem 2.2 by adapting the concavity method (see Levine [20]). We recall the following lemma in [10].

**Lemma 4.1** [10]

*Assume that*${u}_{0}\in {W}_{2}$

*and*$E(0)<d$,

*then it holds that*

*Proof of Theorem 2.2*Assume by contradiction that the solution is global. Then, for any $T>0$, we consider the function $H(\cdot ):[0,T]\to {R}^{+}$ defined by

*T*,

*ρ*are positive constants which will be fixed later. Direct computations show that

*ρ*can be chosen since $E(0)<\frac{(\theta -2)d}{\theta}$), and then

Then we complete the proof by the standard concavity method (Lemma 2.3) since $\theta >2$. □

## Declarations

### Acknowledgements

We thank the referees for their valuable suggestions which helped us improve the paper so much. This work was supported by the National Natural Science Foundation of China (11171311).

## Authors’ Affiliations

## References

- Tan Z: The reaction-diffusion equation with Lewis function and critical Sobolev exponent.
*J. Math. Anal. Appl.*2002, 2: 480-495.View ArticleGoogle Scholar - Pao CV:
*Nonlinear Parabolic and Elliptic Equations*. Plenum, New York; 1992.Google Scholar - Frank-Kamenetskii DA:
*Diffusion and Heat Transfer in Chemical Kinetics*. Plenum, New York; 1969.Google Scholar - Gel’fand IM: Some problems in the theory of quasilinear equations.
*Usp. Mat. Nauk*1959, 2: 87-158. English transl.: Am. Math. Soc. Transl. (2) 29, 295-381 (1963)Google Scholar - Fujuita H:On the nonlinear equations $\delta u+{e}^{u}=0$ and $\partial v/\partial t=\delta v+{e}^{v}$.
*Bull. Am. Math. Soc.*1969, 75: 132-135. 10.1090/S0002-9904-1969-12175-0View ArticleGoogle Scholar - Peral I, Vazquez JL: On the stability or instability of the singular solution of the semilinear heat equation with exponential reaction term.
*Arch. Ration. Mech. Anal.*1995, 129: 201-224. 10.1007/BF00383673MathSciNetView ArticleGoogle Scholar - Pulkkinen A: Blow-up profiles of solutions for the exponential reaction-diffusion equation.
*Math. Methods Appl. Sci.*2011, 34: 2011-2030. 10.1002/mma.1501MathSciNetView ArticleGoogle Scholar - Tello J: Stability of steady states of the Cauchy problem for the exponential reaction-diffusion equation.
*J. Math. Anal. Appl.*2006, 324: 381-396. 10.1016/j.jmaa.2005.12.011MathSciNetView ArticleGoogle Scholar - Ioku N: The Cauchy problem for heat equations with exponential nonlinearity.
*J. Differ. Equ.*2011, 251: 1172-1194. 10.1016/j.jde.2011.02.015MathSciNetView ArticleGoogle Scholar - Alves CO, Cavalcanti MM: On existence, uniform decay rates and blow up for solutions of the 2-d wave equation with exponential source.
*Calc. Var.*2009, 34: 377-411. 10.1007/s00526-008-0188-zMathSciNetView ArticleGoogle Scholar - Sattiger DH: On global solutions of nonlinear hyperbolic equations.
*Arch. Ration. Mech. Anal.*1968, 30: 148-172.Google Scholar - Tsutsumi M: Existence and nonexistence of global solutions for nonlinear parabolic equations.
*Publ. Res. Inst. Math. Sci.*1972/73, 8: 211-229. 10.2977/prims/1195193108MathSciNetView ArticleGoogle Scholar - Iekhata R, Suzuki T: Stable and unstable sets for evolution equations of parabolic and hyperbolic type.
*Hiroshima Math. J.*1996, 26: 475-491.MathSciNetGoogle Scholar - Levine HA, Park SR, Serrin J: Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type.
*J. Differ. Equ.*1998, 142: 212-229. 10.1006/jdeq.1997.3362MathSciNetView ArticleGoogle Scholar - Messaoudi SA: A decay result for a quasilinear parabolic system type. Progress in Nonlinear Differential Equations and Their Applications 63.
*Elliptic and Parabolic Problems*2005, 43-50.Google Scholar - Chen H, Liu GW: Global existence and nonexistence theorems for semilinear parabolic equations with conical degeneration.
*J. Pseud.-Differ. Oper. Appl.*2012, 3(3):329-349. 10.1007/s11868-012-0046-9View ArticleGoogle Scholar - Moser J: A sharp form of an inequality by N. Trudinger.
*Indiana Univ. Math. J.*1971, 20: 1077-1092. 10.1512/iumj.1971.20.20101View ArticleGoogle Scholar - Trudinger NS: On imbeddings into Orlicz spaces and some applications.
*J. Math. Mech.*1967, 17: 473-483.MathSciNetGoogle Scholar - Nakao M, Ono K: Global existence to the Cauchy problem of the semi-linear evolution equations with a nonlinear dissipation.
*Funkc. Ekvacioj*1995, 38: 417-431.MathSciNetGoogle Scholar - Levine HA:Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $p{u}_{t}=-au+f(u)$.
*Arch. Ration. Mech. Anal.*1973, 51: 371-386.View ArticleGoogle Scholar

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