Research Article Reaction-Diffusion in Nonsmooth and Closed Domains

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.