Heat equation with dynamical boundary conditions of locally reactive type

The aim of this paper is to study the well--posedness of the initial-boundary value problem u_t- Delta u=0 in (0,infty) x Omega u_nu=0 on (0,infty)x Gamma_0, u_t=-k_1 u_nu on (0,infty)x Gamma_1, u_t=k_2 u_nu on (0,infty)x Gamma_2, u(0,x)=u_0(x) in \Omega, where Omega is a bounded regular open domain in \R^N (N>=1), Gamma is the boundary of Omega, nu is the outward normal to Omega k_1, k_2>0, Gamma is the union of Gamma_0, Gamma_1 and 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 allows to study the more general initial-boundary value problem u_t-\Delta u=0 in (0,infty) x Omega u_nu=sigma(x) u_t on [0,infty) x Gamma, u(0,x)=u_0(x) on \Omega, where Omega is as before and sigma is in L^infty(Gamma). A key step in our analysis consists in studying a related eigenvalue problem.