Sup-norm estimates for parabolic systems with dynamic boundary conditions

TL;DR

This paper derives rigorous estimates for solutions of parabolic systems with nonlinear dynamic boundary conditions, providing physical interpretations and applicable to biological and ecological models.

Contribution

It introduces new strategies to obtain uniform $L^p$ and $L^ fty$ estimates for such systems, with detailed derivations and physical insights.

Findings

Established $L^p$ and $L^ Infinity$ bounds for solutions

Provided physical interpretations for the boundary conditions

Applied results to models in biology and ecology

Abstract

We consider parabolic systems with nonlinear dynamic boundary conditions, for which we give a rigorous derivation. Then, we give them several physical interpretations which includes an interpretation for the porous-medium equation, and for certain reaction-diffusion systems that occur in mathematical biology and ecology. We devise several strategies which imply (uniform)}$L^{p} and}$L^{\infty}$ estimates on the solutions for the initial value problems considered.