Existence and Stability of Solutions to Nonlocal Parabolic Problems with Variable Exponent.

We are studying a nonlinear nonlocal Dirichlet parabolic problem associated with the following p(x)-laplacian equation ∂u/∂t - a(∫Ω |∇u|p(x)/p(x)) Δ p(x)u = f(x,t). We apply Galerkin’s approximation technique, some uniform estimates and a compactness arguments to prove the existence result. Then we investigate the uniqueness of solutions, finally we give an answer to the asymptotic behavior question. [ABSTRACT FROM AUTHOR]