On the critical exponent for reaction-diffusion equations

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.