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A global approach to fully nonlinear parabolic problems

Authors:	Athanassios G Kartsatos Igor V Skrypnik

Institution:	Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700 ; Institute for Applied Mathematics and Mechanics, R. Luxemburg Str. 74, Donetsk 340114, Ukraine

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 $\begin{displaymath}f\in W^{(2),0}_{p}(Q_{T}),~g \in W^{(3-\frac{1}{p}),0}_{p}(S_{T}). \end{displaymath}$ The resulting problem is then reduced to the problem 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 are achieved via topological methods developed by I. V. Skrypnik. Further applications are also given involving relevant coercive problems, as well as Galerkin approximations.

Keywords:	Initial-boundary value problem mapping of type $(S)_{+} $ Skrypnik's degree theory for demicontinuous mappings of type $(S)_{+} $ Galerkin approximation

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