Boundary Value Problems
Volume 2007 (2007), Article ID 31261, 28 pages
doi:10.1155/2007/31261
Research Article
Reaction-Diffusion in Nonsmooth and Closed Domains
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne 32901, FL, USA
Received 31 May 2006; Revised 6 September 2006; Accepted 21 September 2006
Academic Editor: Vincenzo Vespri
Copyright © 2007 Ugur G. 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.
Abstract
We investigate the Dirichlet problem for the parabolic equation ut=Δum−buβ, m>0, β>0, b∈ℝ, in a nonsmooth and closed domain Ω⊂ℝN+1, N≥2, 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.