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

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 ${\partial }_{t}w\ge 0$. 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).