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Zbl 0822.35068
Escobedo, Miguel; Levine, Howard A.
Critical blowup and global existence numbers for a weakly coupled system of reaction-diffusion equations.
(English)
[J] Arch. Ration. Mech. Anal. 129, No.1, 47-100 (1995). ISSN 0003-9527; ISSN 1432-0673/e

Summary: Let $D\subset \bbfR\sp N$ be either all of $\bbfR\sp N$ or else a cone in $\bbfR\sp N$ whose vertex we may take to be at the origin, without loss of generality. Let $p\sb i$, $q\sb j$, $i=1,2$, be nonnegative with $0<p\sb 1+ q\sb 1\leq p\sb 2+ q\sb 2$. We consider the long-time behavior of nonnegative solutions of the system $$u\sb t= \Delta u+ u\sp{p\sb 1} v\sp{q\sb 1}, \qquad v\sb t= \Delta v+ u\sp{p\sb 2} v\sp{q\sb 2} \tag S$$ in $D\times [0,\infty)$ with $u\sb 0= v\sb 0=0$ on $\partial D$, $(u, v)\sp t (x,0)= (v\sb 0, v\sb 0)\sp t (x)$, $u\sb 0, v\sb 0\geq 0$, $u\sb 0, v\sb 0\in L\sp \infty (D)$. \par We obtain Fujita-type global existence-global non-existence theorems for (S) analogous to the classical result of {\it H. Fujita} [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.340)] and others for the initial-value problem for $u\sb t= \Delta u+ u\sp p$, $u(x,0)= u\sb 0 (x)\geq 0$. The principal result in the case $D= \bbfR\sp N$ and $p\sb 2 q\sb 1>0$ is that when $p\sb 1\geq 1$, the system behaves like the single equation $u\sb t= \Delta u+ u\sp{p\sb 1+ q\sb 1}$ with respect to Fujita- type blowup theorems, whereas if $p\sb 1<1$, the behavior of the system is more complicated. Some of the results extend those of {\it M. Escobedo} and {\it M. A. Herrero} [J. Differ. Equations 89, No. 1, 176- 202 (1991; Zbl 0735.35013)] when $D= \bbfR\sp N$ and of {\it H. A. Levine} and {\it 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 $p\sb 1= q\sb 2 =0$. An example of nonuniqueness is also given.
MSC 2000:
*35K57 Reaction-diffusion equations
35B40 Asymptotic behavior of solutions of PDE
35K40 Systems of parabolic equations, general

Keywords: critical blowup; global existence numbers; global nonexistence; nonuniqueness

Citations: Zbl 0163.340; Zbl 0735.35013; Zbl 0696.35013

Cited in: Zbl 0993.60068 Zbl 0990.35057

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