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Zbl 0913.35065
Mochizuki, Kiyoshi; Huang, Qing
Existence and behavior of solutions for a weakly coupled system of reaction-diffusion equations. (English)
[J] Methods Appl. Anal. 5, No.2, 109-124 (1998). ISSN 1073-2772

Summary: We consider the weakly coupled system of reaction-diffusion equations $$u_t=\Delta u+| x|^{\sigma_1} v^p,\quad v_t=\Delta v+| x|^{\sigma_2}u^q,$$ where $x\in \bbfR^N$ $(N\ge 1)$, $t> 0$, $p,q\ge 1$ with $pq> 1$ and $0\le \sigma_1< N(p- 1)$, $0\le \sigma_2< N(q- 1)$. Put $$\alpha= {2(p+ 1)\over pq-1},\quad \beta= {2(q+ 1)\over pq-1},\quad \delta_1= {\sigma_2p+ \sigma_1\over pq-1},\quad \delta_2= {\sigma_1q+ \sigma_2\over pq-1},$$ and let $I^a$ and $I_a$ $(a\ge 0)$ be the spaces of nonnegative, bounded continuous functions satisfying $$\limsup_{| x|\to\infty} | x|^a\xi(x)< \infty\quad\text{and} \quad \liminf_{| x|\to\infty} | x|^a\xi(x)> 0,$$ respectively. At $t= 0$, initial values $(u_0, v_0)\in I^{\delta_1}\times I^{\delta_2}$ are prescribed. It is proved that if $\max\{\alpha+ \delta_1, \beta+\delta_2\}\ge N$ or if $u_0\in I_a$ with $a< \alpha+\delta_1$ or $v_0\in I_b$ with $b< \beta+\delta_2$, then every nontrivial nonnegative solution is not global in time; whereas if $\max\{\alpha+ \delta_1, \beta+\delta_2\}< N$ and $(u_0, v_0)\in I^a\times I^b$ with $a> \alpha+ \delta_1$, $b> \beta+\delta_2$, then there exist both global solutions and nonglobal solutions. Moreover, we obtain the asymptotic behavior as $t\to\infty$ of the global solutions.

MSC 2000:: *35K57 Reaction-diffusion equations
35B05 General behavior of solutions of PDE
35B40 Asymptotic behavior of solutions of PDE
35K45 Systems of parabolic equations, initial value problems

Keywords: global solutions and nonglobal solutions

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