On critical exponents for the heat equation with a nonlinear boundary condition

Abstract

In this paper we consider the heat equation ut = ∆u in a unbounded domain Ω ⊂ RN with a Neumann boundary condition uv = up, where p > 1 and v is the exterior unit normal on ∂Ω. It is shown for various type of domains that there exists a critical number pc (Ω) ≥ 1, such that all of positive solutions blow up in a finite time when p ∈ (1, pc] while there exist positive global solutions if p > pc and initial data are small.