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Transactions of the American Mathematical Society

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A global approach to fully nonlinear parabolic problems

Authors: Athanassios G. Kartsatos and Igor V. Skrypnik
Journal: Trans. Amer. Math. Soc. 352 (2000), 4603-4640
MSC (1991): Primary 35K55; Secondary 35K30, 35K35
Published electronically: June 13, 2000
MathSciNet review: 1694294
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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 $Au=0,$ 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.

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Additional Information

Athanassios G. Kartsatos
Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

Igor V. Skrypnik
Affiliation: Institute for Applied Mathematics and Mechanics, R. Luxemburg Str. 74, Donetsk 340114, Ukraine

Keywords: Initial-boundary value problem, mapping of type $(S)_{+},$ Skrypnik's degree theory for demicontinuous mappings of type $(S)_{+},$ Galerkin approximation
Received by editor(s): April 18, 1997
Received by editor(s) in revised form: May 7, 1998
Published electronically: June 13, 2000
Additional Notes: This research was partially supported by an NSF-NRC COBASE grant.
Article copyright: © Copyright 2000 American Mathematical Society