Energy stability for a class of two-dimensional interface linear parabolic problems

We consider parabolic equations in two-dimensions with interfaces corresponding to concentrated heat capacity and singular own source. We give an analysis for energy stability of the solutions based on special Sobolev spaces (the energies also are given by the norms of these spaces) that are intrinsic to such problems. In order to define these spaces we study nonstandard spectral problems in which the eigenvalue appears in the interfaces (conjugation conditions) or at the boundary of the spatial domain. The introducing of appropriate spectral problems enable us to precise the values of the parameters which control the energy decay. In fact, in order for numerical calculation to be carried out effectively for large time, we need to know quantitatively this decay property.