Hölder regularity of parabolic equations with Dirichlet boundary conditions and application to reaction-diffusion and reaction-cross-diffusion systems

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 aₜ w - Δw = f with a rough coefficient a, homogeneous Dirichlet boundary conditions, and the special assumption ₜw 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).