Abstract
In this work we consider the energy decay result and nonexistence of global solution for a reactiondiffusion equation with generalized Lewis function and nonlinear exponential growth. There are very few works on the reactiondiffusion equation with exponential growth f as a reaction term by potential well theory. The ingredients used are essentially the TrudingerMoser inequality.
Keywords:
reactiondiffusion equation; stable and unstable set; exponential reaction term; decay rate; global nonexistence1 Introduction
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 reactiondiffusion 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 [2]. The author of work [1] proved the existence and asymptotic estimates of global solutions and finite time blowup 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 [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 blowup of the solution. Tello [8] and Ioku [9] considered the Cauchy problem of heat equation with for .
Recently, Alves and Cavalcanti [10] were concerned with the nonlinear damped wave equation with exponential source. They proved global existence as well as blowup 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 blowup 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 reactiondiffusion 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 [1216] and a long list of references therein. The ingredients used in our proof are essentially the TrudingerMoser 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 blowup 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 , .
(A2) is a function. The function is increasing in , and for each , there exists a positive constant such that
(A3) For each , and fixed, there exists a positive constant such that
(A4) There exists a positive constant such that
A typical example of functions satisfies (A2)(A4) is , with given , , , and .
Now we define some functional as follows:
then the ‘potential depth’ given by
is a positive constant [10]. Hence, we are able to define stable and unstable sets respectively as follows:
We also need the following lemmas.
Let Ω be a bounded domain in. For all,
and there exist positive constantssuch that
Lemma 2.2[19]
Letbe a nonincreasing and nonnegative function on, such that
then
whereC, ωare positive constants depending onand other known qualities.
Lemma 2.3[20]
Suppose that a positive, twicedifferentiable functionsatisfies onthe inequality
where, then there issuch thatas.
In order to state and prove our main results, we remind that by the embedding theorem there exists a constant depending on p and Ω only such that
By multiplying equation (1) by , integrating over Ω, using integration by parts and , we get
Our main results read as follows.
Theorem 2.1Let (A1)(A4) hold. Assume further thatsatisfies
for some sufficiently smalland. Then there exist positive constantsKandksuch that the energysatisfies the decay estimates for larget
Theorem 2.2Let (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 3.1Under the assumptions of Theorem 2.1, we have, for all, .
Proof Since , then there exists (by continuity) such that
This and (A4) give
So, by (15) we have
We then use (5), the Holder inequality and the embedding theorem to obtain, for each ,
Once , we choose β such that , then, from TrudingerMoser inequality (11),
and therefore, by (16) for and , we have
By virtue of (21) and the definition of , we have
This shows that for all . By repeating this procedure and the fact that , we obtain
This is extended to T. □
Lemma 3.2Under the assumptions of Theorem 2.1, we have, for,
Proof It suffices to rewrite (21) as
Thus (22) follows from (23). □
Proof of Theorem 2.1 We integrate (15) over to obtain
Now we multiply (1) by u and integrate over to arrive at
where . Exploiting (14) and (19), we obtain
Using (7), (23) and (22), we have
Integrating both sides of (27) over and using (26), one can write
By using (15) again, we have , , hence
Inserting (29) in (24) and using (27), we easily have
for a constant depending on , A, θ, η only. We then use Young’s inequality to get from (30) and (24)
By (12) in Lemma 2.2 we then get the results. □
4 Proof of the blowup result
In this section, we shall prove Theorem 2.2 by adapting the concavity method (see Levine [20]). We recall the following lemma in [10].
Lemma 4.1[10]
Assume thatand, then it holds that
Proof of Theorem 2.2 Assume by contradiction that the solution is global. Then, for any , we consider the function defined by
where , T, ρ are positive constants which will be fixed later. Direct computations show that
Then, due to equations (1), (7) and (33), we have
Now we take such that (this ρ can be chosen since ), and then
We also note that
Therefore and are both positive. Since for all and , by the construction of , it is clear that
Thus, for all , from (35), (38) and (39) it follows that
which implies
That is,
Then we complete the proof by the standard concavity method (Lemma 2.3) since . □
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors declare that the study was realized in collaboration with the same responsibility. All authors read and approved the final manuscript.
Acknowledgements
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).
References

Tan, Z: The reactiondiffusion equation with Lewis function and critical Sobolev exponent. J. Math. Anal. Appl.. 2, 480–495 (2002)

Pao, CV: Nonlinear Parabolic and Elliptic Equations, Plenum, New York (1992)

FrankKamenetskii, DA: Diffusion and Heat Transfer in Chemical Kinetics, Plenum, New York (1969)

Gel’fand, IM: Some problems in the theory of quasilinear equations. Usp. Mat. Nauk. 2, 87–158 (1959) English transl.: Am. Math. Soc. Transl. (2) 29, 295381 (1963)

Fujuita, H: On the nonlinear equations and . Bull. Am. Math. Soc.. 75, 132–135 (1969). Publisher Full Text

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.. 129, 201–224 (1995). Publisher Full Text

Pulkkinen, A: Blowup profiles of solutions for the exponential reactiondiffusion equation. Math. Methods Appl. Sci.. 34, 2011–2030 (2011). Publisher Full Text

Tello, J: Stability of steady states of the Cauchy problem for the exponential reactiondiffusion equation. J. Math. Anal. Appl.. 324, 381–396 (2006). Publisher Full Text

Ioku, N: The Cauchy problem for heat equations with exponential nonlinearity. J. Differ. Equ.. 251, 1172–1194 (2011). Publisher Full Text

Alves, CO, Cavalcanti, MM: On existence, uniform decay rates and blow up for solutions of the 2d wave equation with exponential source. Calc. Var.. 34, 377–411 (2009). Publisher Full Text

Sattiger, DH: On global solutions of nonlinear hyperbolic equations. Arch. Ration. Mech. Anal.. 30, 148–172 (1968)

Tsutsumi, M: Existence and nonexistence of global solutions for nonlinear parabolic equations. Publ. Res. Inst. Math. Sci.. 8, 211–229 (1972/73). Publisher Full Text

Iekhata, R, Suzuki, T: Stable and unstable sets for evolution equations of parabolic and hyperbolic type. Hiroshima Math. J.. 26, 475–491 (1996)

Levine, HA, Park, SR, Serrin, J: Global existence and nonexistence theorems for quasilinear evolution equations of formally parabolic type. J. Differ. Equ.. 142, 212–229 (1998). Publisher Full Text

Messaoudi, SA: A decay result for a quasilinear parabolic system type. Elliptic and Parabolic Problems. 43–50 (2005)

Chen, H, Liu, GW: Global existence and nonexistence theorems for semilinear parabolic equations with conical degeneration. J. Pseud.Differ. Oper. Appl.. 3(3), 329–349 (2012). Publisher Full Text

Moser, J: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J.. 20, 1077–1092 (1971). Publisher Full Text

Trudinger, NS: On imbeddings into Orlicz spaces and some applications. J. Math. Mech.. 17, 473–483 (1967)

Nakao, M, Ono, K: Global existence to the Cauchy problem of the semilinear evolution equations with a nonlinear dissipation. Funkc. Ekvacioj. 38, 417–431 (1995)

Levine, HA: Some nonexistence and instability theorems for solutions of formally parabolic equations of the form . Arch. Ration. Mech. Anal.. 51, 371–386 (1973)