Heat Equation with Dynamical Boundary Conditions of Locally Reactive Type

Abstract

The aim of this paper is to study the well-posedness of the initial-boundary value problem $\left\{\begin{array}{@{}l@{\hskip 1em}l@{}} u_t-\Delta u=0 &\mbox{in} Q=(0,\infty)\times\Omega,\\ \\[-9pt] u_\nu=0 &\mbox{on} (0,\infty)\times \Gamma_0,\\ \\[-9pt] u_t=-k_1 u_\nu &\mbox{on} (0,\infty)\times\Gamma_1,\\ \\[-9pt] u_t=k_2 u_\nu &\mbox{on} (0,\infty)\times\Gamma_2,\\ \\[-9pt] u(0,x)=u_0(x) & \mbox{in} \Omega, \end{array}\right.$ where $\Omega$ is a bounded regular open domain in ${\Bbb R}^N (N\ge 1), \Gamma=\partial\Omega, \nu$ is the outward normal to $\Omega, k_1,k_2>0$ and $\Gamma=\Gamma_0\cup \Gamma_1\cup \Gamma_2$ , where $\Gamma_i, i=0,1,2$ are pairwise disjoint measurable subsets of $\Gamma$ with respect to Lebesgue surface measure on $\Gamma$ . The main novelty lies on the reactive dynamical boundary condition imposed on $\Gamma_2$ . The technique makes it possible to study the more general initial-boundary value problem $\left\{\begin{array}{@{}l@{\hskip 1em}l@{}} u_t-\Delta u=0 &\mbox{in} Q=(0,\infty)\times\Omega,\\ \\[-9pt] u_\nu=\sigma(x)u_t &\mbox{on} [0,\infty)\times\Gamma,\\ \\[-9pt] u(0,x)=u_0(x) & \mbox{on} \Omega, \end{array}\right.$ where $\Omega$ is as before and $\sigma\in L^\infty(\Gamma)$ . A key step in our analysis consists in studying the eigenvalue problem $\left\{\begin{array}{@{}l@{\hskip 1em}l@{}} -\Delta u=\lambda u,& \mbox{in} \Omega, \\ \\[-9pt] \sigma \Delta u=u_\nu & \mbox{on} \Gamma. \end{array}\right.$