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.
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Communicated by J. serrin
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Meier, P. On the critical exponent for reaction-diffusion equations. Arch. Rational Mech. Anal. 109, 63–71 (1990). https://doi.org/10.1007/BF00377979
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DOI: https://doi.org/10.1007/BF00377979