A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem (1) partial derivative u/partial derivative t - F(x, t, u D(1)u, D(2)u) = f (x,t), (x,t) is an element of Q(T) = Omega x (0, T), (2) G(x, t, u, D(1)u) = g(x, t), (x, t) is an element of S-T = partial deri vative Omega x (0, T) (3) u(x, 0) = h(x), x is an element of Omega, where Omega is a bounded open set in R-n with sufficiently smooth boundary. The problem (1)-(3) is first reduced to the analogous problem in the space W-p((4),0) (Q(T)) with zero initial condition and f is an element of W-p((2),0) (Q(T)), g is an element of W-p((3-1/p),0) (S- T). The resulting problem is then reduced to the problem Au = 0; where the oper ator A : W-p((4),0)(Q(T)) --> [W-p((4),0)(Q(T))]* satisfies Condition (S)(). This reduction is based on a priori estimates which are developed herein for linear parabolic operators with coefficients in Sobolev spaces. The lo cal and global solvability of the operator equation Au = 0 are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approx imations.