H{ö}lder regularity of parabolic equations with Dirichlet boundary conditions and application to reaction-diffusion and reaction-cross-diffusion systems
Hector Bouton, Laurent Desvillettes, Helge Dietert
TL;DR
This work proves Hölder regularity for parabolic equations with rough coefficients under Dirichlet boundary conditions by combining monotone-time-derivative arguments with Dirichlet heat-kernel estimates. The authors develop interior oscillation-reduction and boundary-decay techniques, yielding explicit $C^{0,\alpha}$ bounds in terms of $f\in L^{p}((0,T];L^{q}(\Omega))$ and initial data, with a refined bound involving $f_+$ and critical exponents $\gamma=2-\frac{2}{p}-\frac{d}{q}$. The resulting Dirichlet theory is then applied to nonlinear mass-dissipative systems, establishing global strong solutions in dimensions $d\le 4$ for (i) a reversible chemistry model and (ii) a triangular SKT cross-diffusion system, thereby extending prior Neumann-boundary results to Dirichlet settings. The work provides explicit constants and a robust framework for obtaining a priori estimates and well-posedness for reaction-diffusion and cross-diffusion systems with Dirichlet data, contributing to the mathematical understanding of complex diffusive phenomena in bounded domains.
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 $a\partial_t w - Δw = f$ 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).