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

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).