A global approach to fully nonlinear parabolic problems

Abstract:

We consider the general initial-boundary value problem (1) $\displaystyle {\frac {\partial u}{\partial t}-F(x,t,u,\mathcal {D}^{1}u, \mathcal {D}^{2}u)=f(x,t),\quad (x,t)\in Q_{T}\equiv \Omega \times (0,T),}$ (2) $\displaystyle {G(x,t,u,\mathcal {D}^{1}u)=g(x,t),\quad (x,t)\in S_{T}\equiv \partial \Omega \times (0,T),}$ (3) $\displaystyle {u(x,0)=h(x),\quad x\in \Omega ,}$ 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 \[ f\in W^{(2),0}_{p}(Q_{T}),~g \in W^{(3-\frac {1}{p}),0}_{p}(S_{T}). \] 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.