Volume-surface systems with sub-quadratic intermediate sum on the surface: Global existence and boundedness

TL;DR

This paper proves the global existence and uniform boundedness of solutions to volume-surface reaction-diffusion systems with sub-quadratic nonlinearities, relevant in biological and physical models, using advanced mathematical techniques.

Contribution

It introduces a novel approach combining Moser iteration, $L^p$-energy methods, and duality to handle sub-quadratic growth in volume-surface systems.

Findings

Solutions exist globally in time.

Solutions are uniformly bounded under mass dissipation.

Applicable to biological, ecological, and fluid mechanics models.

Abstract

The global existence and boundedness of solutions to volume-surface reaction diffusion systems with a mass control condition are investigated. Such systems arise typically in e.g. cell biology, ecology or fluid mechanics, when some concentrations or densities are inside a domain and some others are on its boundary. Comparing to previous works, the difficulty of systems under consideration here is that the nonlinearities on the surface can have a sub-quadratic growth rates in all dimensions. To overcome this, we first use the Moser iteration to get some uniform bounds of the time integration of the solutions. Then by combining these bounds with an $L^p$-energy method and a duality argument, we obtain the global existence of solutions. Moreover, under mass dissipation conditions, the solution is shown to be bounded uniformly in time.