This article is part of the series Harnack's Estimates, Positivity and Local Behavior of Degenerate and Singular Parabolic Equations.

Open Access Research Article

Reaction-Diffusion in Nonsmooth and Closed Domains

Ugur G Abdulla

Author Affiliations

Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

Boundary Value Problems 2007, 2007:031261 doi:10.1155/2007/31261


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2007/1/031261


Received:31 May 2006
Revisions received:6 September 2006
Accepted:21 September 2006
Published:30 November 2006

© 2007 Abdulla

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.

We investigate the Dirichlet problem for the parabolic equation in a nonsmooth and closed domain 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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