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A Fujita type global existence-global nonexistence theorem for a weakly coupled system of reaction-diffusion equations. (English) Zbl 0786.35075

The initial value problems for the weakly coupled system, \[ (S(\mathbb{R}^ N)) \quad u_ t=\Delta u+v^ p,\;v_ t=\Delta v+u^ q,\;(x,t)\in \mathbb{R}^ N \times (0,\infty) \] with \(u(x,0)\), \(v(x,0) \geq 0\), \(x \in \mathbb{R}^ N\), was studied in [M. Escobedo and M. A. Herrero, J. Differ. Equations 89, No. 1, 176-202 (1991; Zbl 0735.35013)]. It was shown that if \[ 1<pq \leq 1+2(\max(p,q)+1)/N \] then \((S(\mathbb{R}^ N))\) has no nontrivial global, positive solutions. However, if \[ pq>1+2(\max(p,q)+1)/N \] then both global, bounded nontrivial solutions which blow up in finite time exist.
The purpose of this article is twofold. First, we give a simpler proof of this result of [ M. Escobedo and M. A. Herrero (loc. cit.)] in the special case \(p>1\), \(q \geq 1\). Second, we obtain a similar result for the case when \(\mathbb{R}^ N\) is replaced by a cone and when \(\mathbb{R}^ N\) is replaced by the exterior of a bounded domain. For both geometries, we assume that \(u\), \(v\) vanish on the finite part of the boundary.

MSC:

35K57 Reaction-diffusion equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B40 Asymptotic behavior of solutions to PDEs

Citations:

Zbl 0735.35013
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References:

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