A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations
- Cite this article as:
- Levine, H.A. Z. angew. Math. Phys. (1991) 42: 408. doi:10.1007/BF00945712
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LetD⊂RN be a region with smooth boundary∂D. Letp·q>1,p, q≥1. We consider the system:ut=Δu+vp,vt=Δu+uq inD×[0, ∞) withu=v=0 in∂D×[0, ∞) andu0,v0 nonnegative. Letγ=max(p, q). We show that ifD isRN, 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=RN, (a) holds with equality. An explicit formula forpc(D) is given.