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:

$a(x,t){u}_{t}-\mathrm{\Delta}u=f(u),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},t>0,$

(1)

$u(x,t)=0,\phantom{\rule{1em}{0ex}}x\in \partial \mathrm{\Omega},t>0,$

(2)

$u(x,0)={u}_{0}(x),\phantom{\rule{1em}{0ex}}x\in \mathrm{\Omega},$

(3)

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.