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Properties of positive solutions for a nonlocal reaction-diffusion equation with nonlocal nonlinear boundary condition. (English) Zbl 1203.35150

Summary: We study properties of positive solutions for the reaction-diffusion equation \(u_t = \Delta u + \int _{\Omega }u^p dx - ku^q\) in \(\Omega \times (0,T)\) with nonlocal nonlinear boundary condition \(u(x,t) = \int _{\Omega }f(x,y)u^l (y,t) \,dy\) on \(\partial \Omega \times (0,T)\) and nonnegative initial data \(u_0(x)\), where \(p, q, k, l>0\). Some conditions for the existence and nonexistence of global positive solutions are given.

MSC:

35K61 Nonlinear initial, boundary and initial-boundary value problems for nonlinear parabolic equations
35B35 Stability in context of PDEs
35K57 Reaction-diffusion equations
35K65 Degenerate parabolic equations
35R09 Integro-partial differential equations
35B44 Blow-up in context of PDEs
35B09 Positive solutions to PDEs
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