Weak solutions of the porous medium equation in a cylinder

Abstract:

We show that if $D \subset {{\mathbf {R}}^n}$ is a bounded domain with smooth boundary, and $u \in {L^m}(D \times (\varepsilon ,T))$, $u \geq 0$, solves $\frac {{\partial u}} {{\partial t}} = \Delta {u^m}$, $m > 1$, in the sense of distributions on $D \times (0,T)$, and vanishes on $\partial D \times (0,T)$ in a suitable weak sense, then $u$ is Hölder continuous in $\overline D \times (0,T)$.