In this paper, we study the following initial boundary value problem with generalized Lewis function
which depends on both spacial variable and time:
here is a reaction term with exponential growth at infinity to be specified later, Ω is a bounded domain with smooth boundary ∂ Ω in .
For the reaction-diffusion equation with polynomial growth reaction terms (that is, equation (1) with and ), 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 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 problem (1)-(3) with and the critical Sobolev exponent for .
In this paper we assume that is a reaction term with exponential growth like at infinity. When , , model (1)-(3) 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 for .
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 solutions in finite time by taking the initial data inside the potential well . 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 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 . 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 , , the usual norms in spaces , and , respectively.