A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations

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Abstract

LetDR N be a region with smooth boundary∂D. Letp·q>1,p, q≥1. We consider the system:u tu+v p,v tu+u q inD×[0, ∞) withu=v=0 in∂D×[0, ∞) andu 0,v 0 nonnegative. Letγ=max(p, q). We show that ifD isR N, a cone or the exterior of a bounded domain, then there is a numberpc(D) such that (a) if (γ+1)/(pq−1)>pc(D) no nontrivial global positive solutions of the system exist while (b) if (γ+1)/(pq−1)<pc(D) both nontrivial global and nonglobal solutions exist. In caseD is a cone orD=R N, (a) holds with equality. An explicit formula forpc(D) is given.

This research was supported in part by NSF Grant DMS-8822788 and in part by the Air Force Office of Scientific Research.