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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), $$u_ t-\Delta u=v^ p,\;v_ t-\Delta v=u^ q$$, where $$x\in\mathbb{R}^ N(N\geq1)$$, $$t>0$$, and $$p,q$$ are positive real numbers. At $$t=0$$, nonnegative, continuous and bounded initial values $$(u_ 0(x),v_ 0(x))$$ are prescribed. The corresponding Cauchy problem then has a nonnegative classical and bounded solution $$(u(t,x),v(t,x))$$ in some strip $$S_ T=[0,T)\times\mathbb{R}^ N$$, $$0<T\leq\infty$$. Set $$T^*=\sup\{T>0: u,v \text{ remain bounded in }S_ T\}$$. It is shown in this paper that if $$0<pq\leq1$$, then $$T^*=+\infty$$, so that solutions can be continued for all positive times. When $$pq>1$$ and $$(\gamma+1)/(pq- 1)\geq N/2$$ with $$\gamma=\max\{p,q\}$$, one has $$T^*<+\infty$$ for every nontrivial solution $$(u,v)$$. $$T^*$$ is then called the blow up time of the solution under consideration. Finally, if $$(\gamma+1)(pq-1)<N/2$$ 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 to PDEs 35K57 Reaction-diffusion equations 35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
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