Archive for Rational Mechanics and Analysis

, Volume 109, Issue 1, pp 63–71

On the critical exponent for reaction-diffusion equations

  • Peter Meier
Article

DOI: 10.1007/BF00377979

Cite this article as:
Meier, P. Arch. Rational Mech. Anal. (1990) 109: 63. doi:10.1007/BF00377979

Abstract

In this paper we study the initial-boundary value problem for ut=Δu+ h(t) up with homogeneous Dirichlet boundary conditions, where h(t)tq for large t. Let s* ≔ sup {s ¦ ∃ positive solutions w of ut=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 R2 and R3.

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

Copyright information

© Springer-Verlag 1990

Authors and Affiliations

  • Peter Meier
    • 1
  1. 1.Department of MathematicsIowa State UniversityAmes