Energy decay and nonexistence of solution for a reaction-diffusion equation with exponential nonlinearity
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RESEARCH
Open Access
Energy decay and nonexistence of solution for a reaction-diffusion equation with exponential nonlinearity Huiya Dai1,2* and Hongwei Zhang2 *
Correspondence: [email protected] 1 School of Mathematics and Statistics, Xi’an Jiaotong University, Xianning Road, Xi’an, P.R. China 2 Department of Mathematics, Henan University of Technology, Lianhua Street, Zhengzhou, P.R. China
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: reaction-diffusion equation; stable and unstable set; exponential reaction term; decay rate; global nonexistence
1 Introduction In this paper, we study the following initial boundary value problem with generalized Lewis function a(x, t) which depends on both spacial variable and time: a(x, t)ut – u = f (u), u(x, t) = ,
x ∈ , t > ,
x ∈ ∂, t > ,
u(x, ) = u (x),
x ∈ ,
() () ()
here f (s) is a reaction term with exponential growth at infinity to be specified later, is a bounded domain with smooth boundary ∂ in R . For the reaction-diffusion equation with polynomial growth reaction terms (that is, equation () with a(x, t) = and f (u) = |u|p– u), there have been many works in the literature; one can find a review of previous results in [, ] and references therein, which are not listed in this paper just for concision. Problem ()-() with a(x, t) > describes the chemical reaction processes accompanied by diffusion []. The author of work [] proved the existence and asymptotic estimates of global solutions and finite time blow-up of probn+ for f (u) = up . lem ()-() with a(x, t) > and the critical Sobolev exponent p = n– In this paper we assume that f (s) is a reaction term with exponential growth like es at infinity. When a(x, t) = , f (u) = eu , model ()-() was proposed by [] and []. In this case, Fujita [] studied the asymptotic stability of the solution. Peral and Vazquez [] and Pulkkinen [] considered the stability and blow-up of the solution. Tello [] and Ioku [] considered the Cauchy problem of heat equation with f (u) ≈ eu for |u| ≥ . Recently, Alves and Cavalcanti [] were concerned with the nonlinear damped wave equation with exponential source. They proved global existence as well as blow-up of so© 2014 Dai and Zhang; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Dai and Zhang Boundary Value Problems 2014, 2014:70 http://www.boundaryvalueproblems.com/content/2014/1/70
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