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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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