Bulk-surface systems on evolving domains

TL;DR

This paper investigates bulk-surface systems on evolving domains, establishing global existence, boundedness, and long-term convergence to equilibrium, with applications to biological receptor-ligand dynamics.

Contribution

It provides the first proof of global solutions and boundedness for such systems in all dimensions, and extends the entropy method to evolving domains.

Findings

Global existence and boundedness of solutions in all dimensions.

Solutions converge to a unique equilibrium under certain conditions.

Extended entropy method for evolving bulk-surface systems.

Abstract

Bulk-surface systems on evolving domains are studied. Such problems appear typically from modelling receptor-ligand dynamics in biological cells. Our first main result is the global existence and boundedness of solutions in all dimensions. This is achieved by proving $L^p$-maximal regularity of parabolic equations and duality methods in moving surfaces, which are of independent interest. The second main result is the large time dynamics where we show, under the assumption that the volume/area of the moving domain/surface is unchanged and that the material velocities are decaying for large time, that the solution converges to a unique spatially homogeneous equilibrium. The result is proved by extending the entropy method to bulk-surface systems in evolving domains.