Existence and non-existence of global solutions for a semilinear heat equation

Abstract

The existence and non-existence of global solutions and the Lp blow-up of non-global solutions to the initial value problem u′(t)=Δu(t)+u(t)γ on Rn are studied. We consider only γ>1. In the case n(γ - 1)/2=1, we present a simple proof that there are no non-trivial global non-negative solutions. If n(γ-1)/2≦1, we show under mild technical restrictions that non-negative Lp solutions always blow-up in Lp norm in finite time. In the case n(γ-1)/2>1, we give new sufficient conditions on the initial data which guarantee the existence of global solutions. © 1981 The Weizmann Science Press of Israel.