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A maximum principle for semilinear parabolic systems and applications. (English) Zbl 0986.35044
The authors consider a (Dirichlet) initial-boundary value problem for a semilinear parabolic system \(u_t-\Delta u = f(u,v)\), \(v_t-\Delta v = g(u,v)\) in a bounded subdomain \(\Omega\subset \mathbb{R}^n\). They obtain comparison results for such system in the case when nonlinear terms \(f\) and \(g\) are not Lipschitz. They mostly focus on the nonlinearities \(f(u,v)=u^\alpha v^\beta\), \(g(u,v)=u^\gamma v^\delta\) with positive \(\alpha\), \(\beta\), \(\gamma\), \(\delta\), and describe relations among these parameters for which the mentioned system admits both global and non-global solutions.

MSC:
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K57 Reaction-diffusion equations
35B50 Maximum principles in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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