Homogenization for fully nonlinear parabolic equations

This paper concerns the homogenization of fully nonlinear parabolic equations of the form partial derivative(t)u(epsilon) + H (t, x, t/epsilon(2), x/epsilon, D(2)u(epsilon)) = 0 in (0, T)xR-n, where the Hamiltonian H(t, x, tau, xi, X) is periodic both in tau and xi. Our aim is to establish sufficient conditions for the convergence (as epsilon goes to 0) of u(epsilon) to a solution a to the effective equation partial derivative(t)u + H(t, x, D(2)u) = 0 in (0, T)xR-n, where the effective Hamiltonian H is obtained by a parabolic equation called cell problem. We shall prove that H inherits several properties of H. We also consider the case that: u(epsilon)(0, x) = h (x, x/epsilon) on R-n; we point out a sufficient condition for having u(0, x) = h(x) on R-n,with an effective initial datum h given by the asymptotic behaviour of the solution to the recession problem (a parabolic Cauchy problem related to (1.1)).