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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
##### Keywords:
comparison principles; non-Lipschitz nonlinearity; blow-up
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##### References:
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