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
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.
2 Assumptions and preliminaries
In this section, we present the main assumptions and results. We always assume that:
Now we define some functional as follows:
then the ‘potential depth’ given by
is a positive constant . Hence, we are able to define stable and unstable sets respectively as follows:
We also need the following lemmas.
Our main results read as follows.
3 Proof of decay of the energy
In this section we prove Theorem 2.1. We divide the proof into two lemmas.
This and (A4) give
So, by (15) we have
This is extended to T. □
Proof It suffices to rewrite (21) as
Thus (22) follows from (23). □
Using (7), (23) and (22), we have
Inserting (29) in (24) and using (27), we easily have
By (12) in Lemma 2.2 we then get the results. □
4 Proof of the blow-up result
Then, due to equations (1), (7) and (33), we have
We also note that
The authors declare that they have no competing interests.
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
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).
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