Existence, Uniqueness and Blow-up Result of Solutions for an Evolution p ( x ) (cid:0) laplacian Equation

In this paper we are investigate in the evolution equation p(x)- laplacian with the initial boundary value question. We translate the parabolic equation into the elliptic equation by using a ﬁnite difference method, and then the existence and uniqueness solution are obtained. The blow-up property is shown, by using the energy method. We perform, using Matlab (Ode45 subroutine), some numerical experiments just to illustrate our general results.


INTRODUCTION
Let Ω ⊂ R N (N ≥ 1) be a bounded Lipshitz domain and 0 < T < ∞. It will be assumed throughout this paper that p(x) is continuous function defined in Ω with logarithmic module of continuity : for any x, y ∈ Ω with |x − y| < 1 2 . (1.1) In this paper, we consider the following p(x) − laplacian equation : in Ω, (1.2) where p(x) ∈ C(Ω) is a function. The operator is called p(x)−Laplacian, which will be reduced to the p − Laplacian when p(x) = p a constant.
In the case when a(x) = 1 and p(x) is constant, there have been many results about the existence, uniqueness, and some other properties of the solutions to problem (1.2), we refer to the readers to the bibliography given in [1,2,3] (see also Refs. [4,5,6,7]), and the references therein.
Recently, [8]  In recent years, the research of nonlinear problems with variable exponent growth conditions has been an interesting topic. p(·)-growth problems can be regarded as a kind of nonstandard growth problems and these problems possess very complicated nonlinearities, for instance, the p(x)-Laplacian operator is inhomogeneous. And these problems have many important applications in nonlinear elastic, electrorheological fluids and image restoration.
In this paper, we consider the existence and uniqueness for the problem of the type (1.2) under some assumptions. The proof consists of two steps. First, we prove that the approximating problem admits a global solution; then we do some uniform estimates for these solutions. We mainly use skills of inequality estimation and the method of approximation solutions. By a standard limiting process, we obtain the existence to problem of the type (1.2).
The outline of this paper is the following: In Section 2, we introduce some basic Lebesgue and Sobolev spaces and state our main theorems. In Section 3, we give the existence and uniqueness of weak solutions. In section 4, the blow-up results will be proved. In section 5, we show some numerical experiments.

BASIC SPACES AND THE MAIN RESULTS
To consider problems with variable exponents, one needs the basic theory of spaces L p(x) (Ω) and W 1,p(x) (Ω). For the convenience of readers, let us review them briefly here. The détails and more properties of variable-exponent Lebesgue-Sobolev spaces can be found in [2,22] .
Let p(x) ∈ C(Ω). When p − > 1, one can introduce the variable-exponent Lebesgue space endowed with the Luxemburg norm.
The conjugate space is L q(x) (Ω), with 1 p(x) + 1 q(x) = 1 ∀x ∈ Ω. As in the case of a constant exponent, set endowed with the norm Similarly we also denote by W (Ω) with respect to the inner product in L 2 (Ω).
In Propositions 2.1-2.3, we describe some results about the Luxembourg norm.
Problem (1.2) does not admit classical solutions in general. So, we introduce weak solutions in the following sence.

Definition 2.1.
A function u is said to be a weak solution of Problem (1.2), if the following conditions are satisfied : (Ω)) such that : In the study of the global existence of solutions, we need the following hypotheses (H):

MAIN RESULTS
In this paper, we shall denote by c, Ci differents constants, depending on pi(x), T, Ω, but not on n, which may vary from line to line. Sometimes we shall refer to a constant depending on specific parameters Ci(T ), etc.
Our main existence result is the following: Proof of the main results.

Existence
We will semi-discrete (1.2) in time and solve the corresponding elliptic problem. Based on the semi-discrete problem, we construct the corresponding approximate solutions. The key procedure is to establish necessary a priori estimates for finding the limit of the approximate solutions via a compactess argument.
We first consider the discrete scheme (3.1) where N τ = T and T is a fixed positive real, and 1 ≤ n ≤ N .
(Ω), we consider the functional where g ∈ L ∞ (Ω) is a known function. Using Young's inequality and Proposition 2.1, there exist constants C1, C2 > 0, such that Since the norm is lower semi-continuous and ∫ Ω gudx is continuous functional, Φ(u) is weakly lower semi-continuous on W 1,p(x) 0 (Ω) and satisfy the coercive condition. From [24] we conclude that there exists u * ∈ W 1,p(x) 0 (Ω), such that and u * is the weak solutions of the Euler equation corresponding to Φ(u), Choosing g = f (x, u n−1 ) + a(x) 1 τ u n−1 , we obtain a weak solution u n of (3.1).
Since |f (x, u0)| ≤ M , we may prove by induction that (3.1) has a solution u n in L ∞ (Ω). We put u 1 := w and for any integer k > 0, we may take (w − M τ ) k + as a test function in (3.2) to get ∫ By the Hôlder inequality and |f (x, u0)| ≤ M , we have ∫ We first establish some energy estimates of uτ , uτ .

Uniqueness Theorem 3.4. Let (H1) to (H3) be satisfied. Then problem (1.2) has a unique solution u in QT .
Proof. Let u and v be solutions of (1.2), we have : Since f (x, .) is locally lipschitz uniformly in Ω, the difference w = u − v satisfies We finally deduce from Gronwall's lemma, Thus, we deduce that u = v.

BLOW-UP RESULTS
In this section, we shall investigate the blow-up properties of solutions to problem (1.2), using energy methods. To this end, we consider the following hypotheses on the data.
(H5) f (x, u) = h(u) and and h is such that : Throughout this section, we define for t ≥ 0, By using Hölder's inequality, we have Thus, it is deduced by combining (4.1) and (4.2) that A direct integration of the above inequality over (0, t) then yields which implies that g(t) blows up at a finite time T * ≤ g 1− α 2 (0)/(k( α 2 − 1)), and so does u.

NUMERICAL COMPUTATION
We solve the problem (1.2) with the term f = λu α (x, t), where α > 1 and the domain Ω is just the real line, that is Ω = [0, 1] . The problem becomes : , For the special discretization, we choose a uniform mesh D h = {xi : 0 = x0 < x1 < ... < xM+1 = 1} (with xi = ih) on Ω and replace We use the function 10 sin(πx) as the initial function, we get the following system of Ode, for We solve the problem (S) with the term MATLAB solvers ode45 and we illustrate our previous results with some numerical experiments which show some of the properties observed for the numerical solutions. In all cases we take the initial data u0(x) = 10. sin(π.x), x ∈ [0, 1]. The other parameters are specified in the graphs: (Figs. 1, 2, 3, 4).

CONCLUSION
In paper [8], the author studies the problem (1.2) with the operator div(u m−1 |Du| λ−1 Du) and the source term null, our main contribution is to generalize this work to the p(x)-laplacian operator with the source term f satisfying the conditions of type (4.0). We have carried out several numerical examples in one dimension with variable a(x) and p(x). In equation (S), the non-linear term f (x, u) describes the non-linear source in the diffusion process. We describe our results in the one-dimensional case. Of course, most physical problems are described in two or three dimensions. The extension to several space dimensions is straightforward.