Parabolic problems with dynamical boundary conditions: eigenvalue expansions and blow up

Abstract

An existence theory for local solutions of a parabolic problem under a dynamical boundary condition $\sigma u_t+u_n=0$ is developed and a spectral representation formula is derived. It relies on the spectral theory of an associated elliptic problem with the eigenvalue parameter both in the equation and the boundary condition. The well-posedness of the parabolic problem holds in some natural space only if the number of negative eigenvalues is finite. This depends on the parameter $\sigma$ in the boundary condition. If $\sigma \ge 0$ the parabolic problem is always well-posed. For $\sigma <0$ it is well-posed only if the space dimension is $1$ and ill-posed in space dimension $\ge 2$. By means of the theory of compact operators the spectrum is analyzed and some qualitative properties of the eigenfunctions are derived. An interesting phenomenon is the “parameter-resonance”, where for a specific parameter-value ${\sigma }_0$ two eigenvalues of the elliptic problem cross. Examples are added for which the eigenvalues and eigenfunctions can be computed explicitly. The last part of the paper deals with blow up of solutions of the parabolic problem with nonlinear positive sources.