We study weak formulations of diffusion problems with “dynamical” boundary conditions where the “spatial” differential operator is uniformly strongly elliptic. Separation of time and spatial variables lead to non-coercive quadratic forms, and we introduce the notion ofJ-coercivityto handle this type of problem. The direct method of Courant is employed to prove the validity of the eigenfunction expansions The eigenfunction expansions are then used to construct series solutions of the underlying evolution equations.