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Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. (English) Zbl 0920.35030

The author investigates the asymptotic behavior of solutions to a semilinear parabolic equation on \(\Omega\times \mathbb{R}^+\) with given initial data and nonlocal boundary condition of the form \(Bu(x,t)=\)
\(\int_\Omega K(x,y)u(t,y)dy\), where \(Bu= \alpha_0\partial u/\partial\nu+ u\) and \(\alpha_0\geq 0\) (nonlocal Dirichlet or Robin condition). Under suitable assumptions on \(K\) and the nonlinearity the solution displays corresponding asymptotic behavior. For \(K\geq 0\) and \(\widehat K(x)= \int_\Omega K(x,y)dy\geq 1\), for instance, the solution can blow up in finite time.
Reviewer: B.Kawohl (Köln)

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35K20 Initial-boundary value problems for second-order parabolic equations
35K57 Reaction-diffusion equations
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References:

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