Sharp regularity for the inhomogeneous porous medium equation

Abstract

We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_t-\mathrm{div}\left(m|u{|}^{m-1}\nabla u\right)=f\in L^{q,r},m>1,$$ are locally Hölder continuous, with exponent $$\gamma =\min \left\{\frac{{\alpha }_0^{-}}{m},\frac{[(2q-n)r-2q]}{q[(mr-(m-1)]}\right\},$$ where ${\alpha }_0$ denotes the optimal Hölder exponent for solutions of the homogeneous case. The proof relies on an approximation lemma and geometric iteration in the appropriate intrinsic scaling.

8 pages, LaTeX;