Open Access

Reaction-Diffusion in Nonsmooth and Closed Domains

Boundary Value Problems20062007:031261

DOI: 10.1155/2007/31261

Received: 31 May 2006

Accepted: 21 September 2006

Published: 30 November 2006

Abstract

We investigate the Dirichlet problem for the parabolic equation https://static-content.springer.com/image/art%3A10.1155%2F2007%2F31261/MediaObjects/13661_2006_Article_642_IEq1_HTML.gif in a nonsmooth and closed domain https://static-content.springer.com/image/art%3A10.1155%2F2007%2F31261/MediaObjects/13661_2006_Article_642_IEq2_HTML.gif possibly formed with irregular surfaces and having a characteristic vertex point. Existence, boundary regularity, uniqueness, and comparison results are established. The main objective of the paper is to express the criteria for the well-posedness in terms of the local modulus of lower semicontinuity of the boundary manifold. The two key problems in that context are the boundary regularity of the weak solution and the question whether any weak solution is at the same time a viscosity solution.

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Authors’ Affiliations

(1)
Department of Mathematical Sciences, Florida Institute of Technology

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Copyright

© Ugur G. Abdulla 2007

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.