H{\"o}lder regularity of parabolic equations with Dirichlet boundary conditions and application to reaction-diffusion and reaction-cross-diffusion systems

TL;DR

This paper establishes Hölder regularity for parabolic equations with rough coefficients under Dirichlet boundary conditions and applies these results to prove existence and estimates for complex reaction-diffusion and cross-diffusion systems.

Contribution

It extends previous regularity results to Dirichlet boundary conditions and applies them to demonstrate existence and bounds for reaction-diffusion and cross-diffusion systems.

Findings

Proved Hölder regularity for parabolic equations with rough coefficients and Dirichlet conditions.

Established existence of global strong solutions for SKT cross-diffusion systems.

Derived estimates for reaction-diffusion systems modeling reversible chemistry.

Abstract

In this work, we adapt our recent article [BDD25] to the setting of Dirichlet boundary conditions. A key part is the study of the parabolic equation $a\partial_t w - \Delta w = f$ with a rough coefficient $a$, homogeneous Dirichlet boundary conditions, and the special assumption $\partial_tw \ge 0$. We then apply it to prove existence of global strong solutions to the triangular Shigesada-Kawasaki-Teramoto (SKT) cross-diffusion system with Lotka-Volterra reaction terms in three dimensions and Dirichlet boundary conditions, and to obtain estimates for solutions to reaction-diffusion systems modeling reversible chemistry (still when Dirichlet boundary conditions are considered).