Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions
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 \(Bu(x,t)=\) \break \(\int_\Omega K(x,y)u(t,y)dy\), 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.
scientific article; zbMATH DE number 1135875
scientific article