BLOW-UP, EXPONENTIAL GROUTH OF SOLUTION FOR A NONLINEAR PARABOLIC EQUATION WITH p ( x ) − LAPLACIAN

. In this paper, we consider the following equation We prove a ﬁnite time blowup result for the solutions in the case ω = 0 and exponential growth in the case ω > 0 , with the negative initial energy in the both case.

In particular, this equation arises from the mathematical description of the reaction-diffusion/ diffusion, heat transfer, population dynamics processus, and so on (see [11]) and references therein). Recently in [1], in the case ω = 0, Agaki proved an existence and blow up result for the initial datum u 0 ∈ L r ().Ôtani [17] studied the existence and the asymptotic behavior of solutions of (1.1) and overcome the difficulties caused by the use of nonmonotone perturbation theory. The quasilinear case, with p = 2, requires a strong restriction on the growth of the forcing term |u| r−2 u, which is caused by the loss of the elliptic estimate for the p−Laplacian operator defined by ∆ p u = div(|∇u| p−2 ∇u) (see [2]).
Alaoui et al [12] considered the following nonlinear heat equation Where Ω is a bounded domain in R n with smooth boundary ∂Ω. Under suitable conditions on r and p and for f = 0, they showed that any solution with nontrivial initial datum blows up in finite time. In the absence of the diffusion term in equation (1.1) when p (x) = p and r (x) = r proved the existence and plow up results have been established by many authors (See [1 − 3, 9, 14, 17]).
We should also point out that Polat [18] established a blow-up result for the solution with vanishing initial energy of the following initial boundary value problem Where m and p are real constants.
In recent years, mush attention has been paid to the study of mathematical models of electro-theological fluids. This models inclode hyperbolic, parapolic or elliptic equations which are nonlinear with respect to the gradient of the thought solution with variable exponents of nonlinearity, (see [4,5,10,15]).
Our objective in this paper is to study: In the section 3, the blow up of the solutions of the problem (1.1) in the case ω = 0, in the section 4, exponential growth of solution when ω > 0.
We define the Lebesque space with a variale exponent p (.) by We next, define the variable-exponent Sobolev space W 1,p(.) (Ω) as follows: This is a Banach space with respect to the norm v W 1,p(. (Ω). Let us note that the space W 1,p(.) (Ω) has a differenet definition in the case of variable exponents.
We define the energy functional associaeted of the problem (1.1)

2)
Theorem 3.1. Let the assumptions of proposition 1, be satisfied and assume that Then the solution of the problem (3.1) , blow up in finite time.

Now, we let
and To prove our result, we first establesh some Lemmas.

6)
and Proof. We multiply the first equation of (3.1) by u t and integratying over the domain Ω, we get Integrating (3.8) over (0, t) , we obtain By (3.2) and (3.9) , we have On the other hand, we have Proof of theorem 1. We have Combining of (3.12) , (3.11) and (3.6) , leads to  (3.14) By combining (3.14) and (3.13) , we obtain A direct integration of (3.15) , then yields

Exponential growth
In this section, we prove that the solution of equation (1.1) exponential growth when ω > 0. Proof. where .
Exploiting the algebric inequality Similarly, This gives Proof. By the same procedure of the proof the Lemma 5, we get then, we have for small to be chosen later.
The time derivative of (4.4) , we obtain By using (1.1) , we get To estimate the last term in the right hand side of (4.5) , by using the following Young's Inequality We conclude So, we chosen δ large sufficient and small enough for that we can find λ 1 , λ 2 > 0, such that Combining with (4.12) and (4.10) , we arrive at G (t) ≥ ηG (t) . (4.13) Finally, a simple integration of (4.13) gives G (t) ≥ G (0) e ηt , ∀t ≥ 0. (4.14) Thus completes the proof.