A global approach to fully nonlinear parabolic problems

We consider the general initial-boundary value problem

(1)         ${\frac{\partial u}{\partial t}-F(x,t,u,\mathcal{D}^{1}u, \mathcal{D}^{2}u)=f(x,t),\quad (x,t)\in Q_{T}\equiv \Omega \times (0,T),}$
(2)         ${G(x,t,u,\mathcal{D}^{1}u)=g(x,t),\quad (x,t)\in S_{T}\equiv \partial\Omega \times (0,T),}$
(3)         ${u(x,0)=h(x),\quad x\in \Omega,}$
where $\Omega$ is a bounded open set in $\mathcal{R}^{n}$ with sufficiently smooth boundary.  The problem (1)-(3) is first reduced to the analogous problem in the space $W^{(4),0}_{p}(Q_{T})$with zero initial condition and

The resulting problem is then reduced to the problem $Au=0,$ where the operator $A:W^{(4),0}_{p}(Q_{T})\to \left [W^{(4),0}_{p}(Q_{T})\right ]^{*}$ 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 local 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 approximations.