tailieunhanh - Báo cáo hóa học: " Research Article Reaction-Diffusion in Nonsmooth and Closed Domains"

v Research Article Reaction-Diffusion in Nonsmooth and Closed Domains | Hindawi Publishing Corporation Boundary Value Problems Volume 2007 Article ID31261 28 pages doi 2007 31261 Research Article Reaction-Diffusion in Nonsmooth and Closed Domains Ugur G. Abdulla Received 31 May 2006 Revised 6 September 2006 Accepted 21 September 2006 Recommended by Vincenzo Vespri We investigate the Dirichlet problem for the parabolic equation ut Aum - but m 0 t 0 b e R in a nonsmooth and closed domain o c 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. 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. 1. Introduction Consider the equation ut Aum - but where u u x t x x1 . xN e RN N 2 t e R A N 1 d2 dx2 m 0 t 0 b e R. Equation is usually called a reaction-diffusion equation. It is a simple model for various physical chemical and biological problems involving diffusion with a source b 0 or absorption b 0 of energy see 1 . In this paper we study the Dirichlet problem DP for in a general domain o c RN 1 with do being a closed N-dimensional manifold. It can be stated as follows given any continuous function on the boundary do of o to find a continuous extension of this function to the closure of o which satisfies in o. 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. 2 Boundary Value