Blowup of Solution for a Class of Doubly Nonlinear Parabolic Systems

where k > 2, p > 2 are real numbers and Ω is bounded domain in Rn with smooth boundary ∂Ω so that the divergence theorem can be applied. Here Δ denotes the Laplace operator inΩ. This type of problems not only is important from the theoretical point of view, but also arises in many physical applications and describes a great deal of models in applied science. It appears in the models of chemical reactions, heat transfer, and population dynamics (see [1] and references therein). Equation (1) can describe an electric breakdown in crystalline semiconductors with allowance for the linear dissipation of boundand free-charge sources [2, 3]. In the absence of the nonlinear diffusion term |u|k−2u t , (1) reduced to the following equation


Introduction
In this paper, we study the following initial boundary value problem of a class of reaction diffusion equations with multiple nonlinearities: (, ) = 0,  ∈ Ω, (, 0) =  0 () ,  ∈ Ω, where  > 2,  > 2 are real numbers and Ω is bounded domain in   with smooth boundary Ω so that the divergence theorem can be applied.Here Δ denotes the Laplace operator in Ω.This type of problems not only is important from the theoretical point of view, but also arises in many physical applications and describes a great deal of models in applied science.It appears in the models of chemical reactions, heat transfer, and population dynamics (see [1] and references therein).Equation (1) can describe an electric breakdown in crystalline semiconductors with allowance for the linear dissipation of bound-and free-charge sources [2,3].
In the absence of the nonlinear diffusion term || −2   , (1) reduced to the following equation which is called pseudoparabolic equation (see [4] and the references).A related problem to (4) without term −Δ  has attracted a great deal of attention in the last two decades, and many results appeared on the existence, blowup, and asymptotic behavior of solution.It is well known that the nonlinear || −2  reaction term drives the solution of (4) to blowup in finite time.The diffusion term is known to yield existence of global solution if the reaction term is removed from [5].The more general equation, has also attracted a great deal of people and the known results show that global existence and nonexistence depend roughly on , the degree of nonlinearity in , the dimension , and the size of the initial data.See, in this regard, the works of Levine [6], Kalantarov and Ladyzhenskaya [7], Levine et al. [8], Messaoudi [9], Liu and Wang [10] and references therein.Pucci and Serrin [11] have discussed the stability of the following equation: Levine et al. [8] got the global existence and nonexistence of solution for (6).Pang et al. [12,13] and Berrimi and Messaoudi [14] gave the sufficient condition of blowup result for certain solutions of ( 6) with positive or negative initial energy. 2
Polat [20] established a blowup result for the solution with vanishing initial energy of the following initial boundary value problem: They also gave detailed results of the necessary and sufficient blowup conditions together with blowup rate estimates for the positive solution of the problem, subject to various boundary conditions.Korpusov and Sveshnikov [2,3] gave the local strong solution and the sufficient close-to-necessary conditions for the blowup of solutions to the problem, with initial boundary values ( 2) and (3) in  3 for ,  > 0 by the convex method [6,7].
In this paper, we will investigate the problem (1)-( 3) and there are few results of the problem to our knowledge.We will give sufficient conditions for the blowup of solutions in a finite time interval under suitable initial data using differential inequalities.An essential tool of the proof is an idea used in [21,22], which was based on an auxiliary function (which is a small perturbation of the total energy), using differential inequalities and obtaining the result.It is different with the result of [2,3].This paper is organized as follows.Section 2 is concerned with some notations and statement of assumptions.In Section 3, we give and prove that the result if the initial energy (0) of our solutions is negative (this means that our initial data are large enough) or the initial energy is (0) > 0.

Preliminaries
In this section, we will give some notations and statement of assumptions for , , .We denote   (Ω) by   ,  1 0 (Ω) by  1 0 , the usual Soblev space.The norm and inner of   (Ω) are denoted by For the numbers  and , we assume that Similar to [2], we call (, ) a solution of problem ( 1) satisfying (, 0) =  0 () and Now, we introduce two functionals: where  ∈  1 0 .Multiplying (1) by   and integrating over Ω, we have and then

Blowup of Solution
In this section, we will prove the main result.Our techniques of proof follow very carefully the techniques used in [21,22].
Theorem 1. Suppose that the assumption about ,  hold,  0 ∈  1 0 and  is a local solution of the system (1)-( 3), and (0) < 0 is sufficient negative.Then the solution of the system (1)- (3) blows up in finite time.
Proof.We set
In the following, we will prove that the energy will grow up as an exponential function as time goes to infinity, provided that the initial energy (0) > 0.
The following lemma will play an essential role in the proof of our main result, and it is similar to a Lemma used firstly by Vitillaro [23].In order to give the result and for the sake of simplicity, we set where  * is the best Poincare's constant.