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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), $u\sb t-\Delta u=v\sp p,\ v\sb t-\Delta v=u\sp q$, where $x\in\bbfR\sp N(N\ge1)$, $t>0$, and $p,q$ are positive real numbers. At $t=0$, nonnegative, continuous and bounded initial values $(u\sb 0(x),v\sb 0(x))$ are prescribed. The corresponding Cauchy problem then has a nonnegative classical and bounded solution $(u(t,x),v(t,x))$ in some strip $S\sb T=[0,T)\times\bbfR\sp N$, $0<T\le\infty$. Set $T\sp*=\sup\{T>0: u,v \text{ remain bounded in }S\sb T\}$. It is shown in this paper that if $0<pq\le1$, then $T\sp*=+\infty$, so that solutions can be continued for all positive times. When $pq>1$ and $(\gamma+1)/(pq- 1)\ge N/2$ with $\gamma=\max\{p,q\}$, one has $T\sp*<+\infty$ for every nontrivial solution $(u,v)$. $T\sp*$ 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 $S\sb T$ while others exhibit finite blow up times.

35B40Asymptotic behavior of solutions of PDE
35K57Reaction-diffusion equations
35B30Dependence of solutions of PDE on initial and boundary data, parameters
Full Text: DOI
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