Optimal global second-order regularity and improved integrability for parabolic equations with variable growth

We consider the homogeneous Dirichlet problem for the parabolic equation u(t)- div(|del u|(p(x,t)-2)del u) = f(x, t) + F (x, t, u, del u) in the cylinder Q(T):= Omega x (0, T), where Omega subset of R-N, N >= 2, is a C(2 )smooth or convex bounded domain. It is assumed that p is an element of C (0,1) (Q(T)) is a given function and that the nonlinear source F (x, t, s, xi) has a proper power growth with respect to s and xi. It is shown that if p(x, t) > 2(N+1)/ N+2 , f is an element of L-2(Q(T)) |del u0|(P(x,0) )is an element of L-1(Omega), then the problem has a solution u is an element of C-0 ([0, T]; L-2(Omega)) with |del u|(p(x,t)) is an element of L-infinity(0, T; L-1(Omega)), u(t) is an element of L-2(Q(T)) obtained as the limit of solutions to the regularized problems in the parabolic H & ouml;lder space. The solution possesses the following global regularity properties: |del u|(2(p(x,t)-1)+r )is an element of L-1(Q(T)), for any 0< r < 4/ N+2, |del u|(p(x,t)-2 )del u is an element of L-2 (0, T; W-1,W-2(Omega))(N).