Homogenization of solutions of initial boundary value problems for parabolic systems

Let O subset of R-d be a bounded C-1,C-1 domain. In L-2(O; C-n) we consider strongly elliptic operators A(D,epsilon) and A(N,epsilon) given by the differential expression b(D)*g(x/epsilon)b(D), epsilon > 0, with Dirichlet and Neumann boundary conditions, respectively. Here g(x) is a bounded positive definite matrix-valued function assumed to be periodic with respect to some lattice and b(D) is a first-order differential operator. We find approximations of the operators exp(-A(D,epsilon)t) and exp(-A(N,epsilon)t) for fixed t > 0 and small epsilon in the L-2 -> L-2 and L-2 -> H-1 operator norms with error estimates depending on epsilon and t. The results are applied to homogenize the solutions of initial boundary value problems for parabolic systems.