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Boundedness and blow up for a semilinear reaction-diffusion system. (English) Zbl 0735.35013
From authors’ abstract: Consider the semilinear parabolic system (S), ${u}_{t}-{\Delta }u={v}^{p},\phantom{\rule{4pt}{0ex}}{v}_{t}-{\Delta }v={u}^{q}$, where $x\in {ℝ}^{N}\left(N\ge 1\right)$, $t>0$, and $p,q$ are positive real numbers. At $t=0$, nonnegative, continuous and bounded initial values $\left({u}_{0}\left(x\right),{v}_{0}\left(x\right)\right)$ are prescribed. The corresponding Cauchy problem then has a nonnegative classical and bounded solution $\left(u\left(t,x\right),v\left(t,x\right)\right)$ in some strip ${S}_{T}=\left[0,T\right)×{ℝ}^{N}$, $0. Set ${T}^{*}=sup\left\{T>0:u,v\phantom{\rule{4.pt}{0ex}}\text{remain}\phantom{\rule{4.pt}{0ex}}\text{bounded}\phantom{\rule{4.pt}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{S}_{T}\right\}$. It is shown in this paper that if $0, then ${T}^{*}=+\infty$, so that solutions can be continued for all positive times. When $pq>1$ and $\left(\gamma +1\right)/\left(pq-1\right)\ge N/2$ with $\gamma =max\left\{p,q\right\}$, one has ${T}^{*}<+\infty$ for every nontrivial solution $\left(u,v\right)$. ${T}^{*}$ is then called the blow up time of the solution under consideration. Finally, if $\left(\gamma +1\right)\left(pq-1\right) both situations coexist, since some nontrivial solutions remain bounded in any strip ${S}_{T}$ while others exhibit finite blow up times.

##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 35K57 Reaction-diffusion equations 35B30 Dependence of solutions of PDE on initial and boundary data, parameters