Energy decay and nonexistence of solution for a reaction-diffusion equation with exponential nonlinearity
© Dai and Zhang; licensee Springer. 2014
Received: 2 December 2013
Accepted: 13 March 2014
Published: 25 March 2014
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.
Keywordsreaction-diffusion equation stable and unstable set exponential reaction term decay rate global nonexistence
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.
2 Assumptions and preliminaries
In this section, we present the main assumptions and results. We always assume that:
(A1) is a positive differentiable function and is bounded for , .
A typical example of functions satisfies (A2)-(A4) is , with given , , , and .
We also need the following lemmas.
Lemma 2.2 
where C, ω are positive constants depending on and other known qualities.
Lemma 2.3 
where , then there is such that as .
Our main results read as follows.
Theorem 2.2 Let (A1)-(A4) hold. Assume further that , and , 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 , .
This is extended to T. □
Thus (22) follows from (23). □
By (12) in Lemma 2.2 we then get the results. □
4 Proof of the blow-up result
Lemma 4.1 
Then we complete the proof by the standard concavity method (Lemma 2.3) since . □
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).
- 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 and . 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 . Arch. Ration. Mech. Anal. 1973, 51: 371-386.View ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. 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 credited.