Lifespan estimates via Neumann heat kernel

We study the lower bound of the lifespan T for the heat equation ut=u in a bounded domain Rn(n2) with positive initial data u0 and a nonlinear radiation condition on partial boundary: the exterior normal derivative u/n=uq on 1 for some q>1, while u/n=0 on the other part of the boundary. Previously, under the convexity assumption of , the asymptotic behaviours of T on the maximum M0 of u0 and the surface area |1| of 1 were explored. This paper is intended to remove the convexity assumption of since it is very restrictive in real applications. We will show that as M00+, T is at least of order M0-(q-1) which is optimal, and meanwhile prove that as |1|0+, T is at least of order |1|-1n-1 for n3 and |1|-1/ln|1|-1 for n=2. The order of T on |1| when n=2 is almost optimal. Instead of the usual energy method, the proofs in this paper are carried out by carefully analysing the representation formula of u in terms of the Neumann heat kernel.