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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Well-posedness of the Dirichlet problem for the non-linear diffusion equation in non-smooth domains


Author: Ugur G. Abdulla
Translated by:
Journal: Trans. Amer. Math. Soc. 357 (2005), 247-265
MSC (2000): Primary 35K65, 35K55
Published electronically: February 27, 2004
MathSciNet review: 2098094
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Abstract: We investigate the Dirichlet problem for the parablic equation

\begin{displaymath}u_t = \Delta u^m, m > 0, \end{displaymath}

in a non-smooth domain $\Omega \subset \mathbb{R}^{N+1}, N \geq 2$. In a recent paper [U.G. Abdulla, J. Math. Anal. Appl., 260, 2 (2001), 384-403] existence and boundary regularity results were established. In this paper we present uniqueness and comparison theorems and results on the continuous dependence of the solution on the initial-boundary data. In particular, we prove $L_1$-contraction estimation in general non-smooth domains.


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Additional Information

Ugur G. Abdulla
Affiliation: Department of Mathematical Sciences, Florida Institute of Technology, 150 West University Boulevard, Melbourne, Florida 32901-6975
Email: abdulla@fit.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03464-6
PII: S 0002-9947(04)03464-6
Keywords: Dirichlet problem, non-smooth domains, non-linear diffusion, degenerate and singular parabolic equations, uniqueness and comparison results, $L_1$-contraction, boundary gradient estimates.
Received by editor(s): July 31, 2000
Received by editor(s) in revised form: July 21, 2003
Published electronically: February 27, 2004
Article copyright: © Copyright 2004 American Mathematical Society