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 partial derivative u/partial derivative t - a(integral(Omega)vertical bar del u vertical bar(p(x))/p(x))Delta(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.