Existence of solution for a class of heat equation involving the p ( x ) Laplacian with triple regime
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Existence of solution for a class of heat equation involving the p(x) Laplacian with triple regime Claudianor O. Alves and Tahir Boudjeriou
Abstract. In this paper, we study the local and global existence of solution and the blow-up phenomena for a class of heat equation involving the p(x)-Laplacian with triple regime. Mathematics Subject Classification. 35K59, 65M60, 35B44. Keywords. Quasilinear parabolic equations, Galerkin Methods, Blow-up.
1. Introduction In this paper, we study the local and global existence of solution for the following class of heat equation ⎧ ⎨ ut − Δp(x) u = |u|q(x)−2 u in Ω × (0, +∞), (1.1) u = 0, on ∂Ω × (0, +∞), ⎩ in Ω, u(x, 0) = u0 (x), 1,p(x)
where Ω ⊆ RN is a smooth bounded domain and u0 ∈ W0,rad (Ω) and p, q : Ω → R are continuous functions satisfying some conditions that will be mentioned later on. When p and q are constant functions, the problem above becomes ut − Δp u = |u|q−2 u in Ω × (0, +∞), (1.2) in Ω. u(x, 0) = u0 (x), The methods used to solve that problem are developed in relationship with the values of q with respect to the Sobolev critical exponent p∗ of p, which is defined by ⎧ ⎨ Np if 1 < p < N p∗ = N −p ⎩ +∞ if p ≥ N, and only one of the situations below can occur: (i) q < p∗ (subcritical case). (ii) q = p∗ , provided that 1 < p < N (critical case). (iii) q > p∗ , provided that 1 < p < N (supercritical case).
C. O. Alves was partially supported by CNPq/Brazil 304804/2017-7. 0123456789().: V,-vol
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C. O. Alves and T. Boudjeriou
ZAMP
In the case of variable exponents, the problem (1.2) becomes more rich, because it can fulfill even a “subcritical–critical–supercritical” triple regime, in the sense that we can have Ω = Ω1 ∪ Ω2 ∪ Ω3 with q(x) < p∗ (x) q(x) = p∗ (x) q(x) > p∗ (x)
if x ∈ Ω1 ; if x ∈ Ω2 ; if x ∈ Ω3 .
(1.3)
In the last few decades, special attention has been paid to the study of partial differential equations involving p(x)-growth conditions. The interest in studying such problems is motivated by their applications in image restoration, nonlinear elasticity theory, electrorheological fluids, and so forth. In particular, parabolic equations involving the p(.)-Laplacian are related to the field of electrorheological fluids which are characterized by their ability to change the mechanical properties under the influence of the exterior electromagnetic; for a more physical motivation we refer the reader to [13,22] and [23] and their references. The rigorous study for these physical problems has been facilitated by the development of Lebesgue and Sobolev spaces with variable exponents. It is important to point out that many results have obtained on parabolic equations with nonlinearities of variable exponent where the authors have studied the global existence in the subcritical case (q < p∗ ), for example, see ([5,6,24,27–30,32]) and the references therein. In [33], by using the Galerkin method, the authors have studied the global existence and asymptotic behavior of global weak solutions
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