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

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