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Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations. (English) Zbl 0822.35068
Summary: Let \(D\subset \mathbb{R}^ N\) be either all of \(\mathbb{R}^ N\) or else a cone in \(\mathbb{R}^ N\) whose vertex we may take to be at the origin, without loss of generality. Let \(p_ i\), \(q_ j\), \(i=1,2\), be nonnegative with \(0<p_ 1+ q_ 1\leq p_ 2+ q_ 2\). We consider the long-time behavior of nonnegative solutions of the system \[ u_ t= \Delta u+ u^{p_ 1} v^{q_ 1}, \qquad v_ t= \Delta v+ u^{p_ 2} v^{q_ 2} \tag{S} \] in \(D\times [0,\infty)\) with \(u_ 0= v_ 0=0\) on \(\partial D\), \((u, v)^ t (x,0)= (v_ 0, v_ 0)^ t (x)\), \(u_ 0, v_ 0\geq 0\), \(u_ 0, v_ 0\in L^ \infty (D)\).
We obtain Fujita-type global existence-global non-existence theorems for (S) analogous to the classical result of H. Fujita [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.340)] and others for the initial-value problem for \(u_ t= \Delta u+ u^ p\), \(u(x,0)= u_ 0 (x)\geq 0\). The principal result in the case \(D= \mathbb{R}^ N\) and \(p_ 2 q_ 1>0\) is that when \(p_ 1\geq 1\), the system behaves like the single equation \(u_ t= \Delta u+ u^{p_ 1+ q_ 1}\) with respect to Fujita- type blowup theorems, whereas if \(p_ 1<1\), the behavior of the system is more complicated. Some of the results extend those of M. Escobedo and M. A. Herrero [J. Differ. Equations 89, No. 1, 176- 202 (1991; Zbl 0735.35013)] when \(D= \mathbb{R}^ N\) and of H. A. Levine and P. Meier [Isr. J. Math. 67, No. 2, 129-136 (1989; Zbl 0696.35013)] when \(D\) is a cone. These authors considered (S) in the case of \(p_ 1= q_ 2 =0\). An example of nonuniqueness is also given.

MSC:
35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions to PDEs
35K40 Second-order parabolic systems
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