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Zbl 0920.35030
Pao, C.V.
Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions.
(English)
[J] J. Comput. Appl. Math. 88, No.1, 225-238 (1998). ISSN 0377-0427

The author investigates the asymptotic behavior of solutions to a semilinear parabolic equation on $\Omega\times \bbfR^+$ with given initial data and nonlocal boundary condition of the form $Bu(x,t)=$ \break $\int_\Omega K(x,y)u(t,y)dy$, where $Bu= \alpha_0\partial u/\partial\nu+ u$ and $\alpha_0\ge 0$ (nonlocal Dirichlet or Robin condition). Under suitable assumptions on $K$ and the nonlinearity the solution displays corresponding asymptotic behavior. For $K\ge 0$ and $\widehat K(x)= \int_\Omega K(x,y)dy\ge 1$, for instance, the solution can blow up in finite time.
[B.Kawohl (Köln)]
MSC 2000:
*35B40 Asymptotic behavior of solutions of PDE
35K20 Second order parabolic equations, boundary value problems
35K57 Reaction-diffusion equations

Keywords: nonlocal Dirichlet or Robin condition; semilinear parabolic equation; blow up in finite time

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