On Critical Exponents for the Heat Equation with a Mixed Nonlinear Dirichlet]Neumann Boundary Condition

In this paper we consider the heat equation utsDu in an unbounded domain V;RN with a partly Dirichlet condition u x, t.s0 and a partly Neumann condition u su p on the boundary, where p)1 and n is the exterior unit normal n on the boundary. It is shown that for a sectorial domain in R2 and an orthant domain in RN there exists an explicit critical exponent pc V.)1 such that all positive solutions blow up in finite time when pg 1, pc x while there exist positive global solutions if p)pc and initial data are suitably small. All our blowup results include the critical case