, Volume 109, Issue 1, pp 63-71

On the critical exponent for reaction-diffusion equations

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Abstract

In this paper we study the initial-boundary value problem for u t =Δu+ h(t) u p with homogeneous Dirichlet boundary conditions, where h(t)t q for large t. Let s * ≔ sup {s ¦ ∃ positive solutions w of u t =Δu such that \(\mathop {\lim \sup }\limits_{t \to \infty } t^s \parallel w( \cdot ,t)\parallel _\infty < \infty \}\) . Then for p * ≔ 1+(q+1)/s * we show: If p>p *, there are global positive solutions that decay to zero uniformly for t→∞. If 1<p<p *, then all nontrivial solutions blow up in finite time. We determine p * for some conical domains in R 2 and R 3.

A similar result is derived for a bounded domain if h(t)e βt for large t.

Communicated by J. serrin