Spectral transition and variational localization in multilayer transport with interface resistance and dissipation

Abstract

We investigate transport in multilayer systems combining heterogeneous diffusion, interfacial resistance, and volumetric dissipation. Unlike classical diffusion models, the presence of dissipative bulk terms and imperfect interfaces leads to a dissipative diffusion–transmission operator structure. We show that the spectral properties of the associated diffusion–efflux operator are governed by a single mechanism: the competition between volumetric dissipation and interfacial transmission. This competition is quantified by a dimensionless parameter that controls both the principal eigenvalue and the spatial structure of the corresponding eigenfunction. In particular, the system exhibits a transition between extended and localized modes, providing a spectral mechanism for transport limitation in heterogeneous media. Variational localization arises as a variational consequence of dissipative contrasts and does not require modification of the constitutive diffusion law. The formulation admits a natural interpretation within the framework of generalized continuum mechanics, where interface terms define a surface energy analogous to micromorphic or Cosserat-type interactions. Numerical experiments confirm the predicted scaling laws and the emergence of variational localization across regimes. These results provide a unified spectral framework for understanding transport in systems with internal interfaces and dissipation.