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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 $D\subset {ℝ}^{N}$ be either all of ${ℝ}^{N}$ or else a cone in ${ℝ}^{N}$ 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 $0<{p}_{1}+{q}_{1}\le {p}_{2}+{q}_{2}$. We consider the long-time behavior of nonnegative solutions of the system

${u}_{t}={\Delta }u+{u}^{{p}_{1}}{v}^{{q}_{1}},\phantom{\rule{2.em}{0ex}}{v}_{t}={\Delta }v+{u}^{{p}_{2}}{v}^{{q}_{2}}\phantom{\rule{2.em}{0ex}}\left(\mathrm{S}\right)$

in $D×\left[0,\infty \right)$ with ${u}_{0}={v}_{0}=0$ on $\partial D$, ${\left(u,v\right)}^{t}\left(x,0\right)={\left({v}_{0},{v}_{0}\right)}^{t}\left(x\right)$, ${u}_{0},{v}_{0}\ge 0$, ${u}_{0},{v}_{0}\in {L}^{\infty }\left(D\right)$.

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 ${u}_{t}={\Delta }u+{u}^{p}$, $u\left(x,0\right)={u}_{0}\left(x\right)\ge 0$. The principal result in the case $D={ℝ}^{N}$ and ${p}_{2}{q}_{1}>0$ is that when ${p}_{1}\ge 1$, 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 $D={ℝ}^{N}$ 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 of PDE 35K40 Systems of second-order parabolic equations, general
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