$L^p(I,C^\alpha(\Omega))$ Regularity for Reaction-Diffusion Equations with Non-smooth Data

TL;DR

This paper establishes regularity results in the space $L^p(I,C^eta(ar{ abla}))$ for reaction-diffusion equations with non-smooth data, providing explicit norm estimates and extending previous stationary regularity results.

Contribution

It introduces new regularity estimates for reaction-diffusion equations with non-smooth data in mixed boundary conditions, extending prior stationary regularity results to the time-dependent setting.

Findings

Proves $L^p(I,C^eta(ar{ abla}))$ regularity for reaction-diffusion equations.

Provides explicit bounds of the solution norm in terms of data.

Extends stationary regularity results to evolution equations.

Abstract

We prove an $L^p(I,C^\alpha(\Omega))$ regularity result for a reaction-diffusion equation with mixed boundary conditions, symmetric $L^\infty$ coefficients and an $L^\infty$ initial condition. We provide explicit control of the $L^p(I,C^\alpha(\Omega))$ norm with respect to the data. To prove our result, we first establish $C^\alpha(\Omega)$ control of the stationary equation, extending a result by Haller-Dintelmann et al. (2009).