Global well-posedness for volume-surface reaction-diffusion systems

TL;DR

This paper proves the global existence of classical solutions for volume-surface reaction-diffusion systems with mass control, introducing new energy methods and conditions to handle the complex volume-surface coupling.

Contribution

It develops a novel $L^p$-energy approach combined with a duality method to establish global solutions under the intermediate sum condition and quasi-uniform diffusion assumptions.

Findings

Global existence of classical solutions under the intermediate sum condition.

Solutions are uniformly bounded in time for mass-dissipative systems.

Applicable to biological models like membrane protein clustering and cell polarization.

Abstract

We study the global existence of classical solutions to volume-surface reaction-diffusion systems with control of mass. Such systems appear naturally from modeling evolution of concentrations or densities appearing both in a volume domain and its surface, and therefore have attracted considerable attention. Due to the characteristic volume-surface coupling, global existence of solutions to general systems is a challenging issue. In particular, the duality method, which is powerful in dealing with mass conserved systems in domains, is not applicable on its own. In this paper, we introduce a new family of $L^p$-energy functions and combine them with a suitable duality method for volume-surface systems, to ultimately obtain global existence of classical solutions under a general assumption called the \textit{intermediate sum condition}. For systems that conserve mass, but do not satisfy this condition, global solutions are shown under a quasi-uniform condition, that is, under the assumption that the diffusion coefficients are close to each other. In the case of mass dissipation, we also show that the solution is bounded uniformly in time by studying the system on each time-space cylinder of unit size, and showing that the solution is sup-norm bounded independently of the cylinder. Applications of our results include global existence and boundedness for systems arising from membrane protein clustering or activation of Cdc42 in cell polarization.