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

DOI:
https://doi.org/10.1090/S0002-9947-00-02541-1

Published electronically:
June 13, 2000

MathSciNet review:
1694294

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Abstract | References | Similar Articles | Additional Information

Abstract: We consider the general initial-boundary value problem

(1)

(2)

(3)

where is a bounded open set in with sufficiently smooth boundary. The problem (1)-(3) is first reduced to the analogous problem in the space with zero initial condition and

The resulting problem is then reduced to the problem where the operator satisfies Condition 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.

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Additional Information

**Athanassios G. Kartsatos**

Affiliation:
Department of Mathematics, University of South Florida, Tampa, Florida 33620-5700

Email:
hermes@math.usf.edu

**Igor V. Skrypnik**

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

Email:
skrypnik@iamm.ac.donetsk.ua

DOI:
https://doi.org/10.1090/S0002-9947-00-02541-1

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