Global existence, uniqueness and stability for nonlinear dissipative systems of bulk-interface interaction

Abstract

We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and e.g. slow, fast and entropic diffusion processes under mass conservation. The main results are global well-posedness and exponential stability of equilibria. As a part of the proof, we show bulk-interface maximum principles and a bulk-interface Poincaré inequality. The method of proof for global existence is a simple but very versatile combination of maximal parabolic regularity of the linearization, a priori L1-bounds and a Schaefer fixed point argument. This allows us to extend the setting e.g. conditions and external forces.