REACTION-DIFFUSION EQUATIONS WITH SPATIALLY VARIABLE EXPONENTS AND LARGE DIFFUSION

In this work we prove continuity of solutions with respect to initial conditions and couple parameters and we prove joint upper semicontinuity of a family of global attractors for the problem {partial derivative u(s)/partial derivative t(t) - div(D-s vertical bar del u(s)vertical bar(ps(x)-2)del u(s)) + vertical bar u(s)vertical bar(ps(x)-2)u(s) = B(u(s)(t)), t > 0, u(s)(0) = u(0s), under homogeneous Neumann boundary conditions, u0(s) epsilon H := L-2(Omega), Omega subset of R-n (n >= 1) is a smooth bounded domain, B : H -> H is a globally Lipschitz map with Lipschitz constant L >= 0, D-s epsilon [1, infinity), p(s)(.) epsilon C((Omega) over bar), (p(s)) over bar :=ess inf p(s) >= p, p(s)(+) :=ess sup P-s <= a, for all s epsilon N, when p(s)(.) -> p in L infinity(Omega) and D-s -> infinity as s -> infinity, with a, p > 2 positive constants.