Sharp regularity for the inhomogeneous porous medium equation

TL;DR

This paper proves local Hölder continuity for solutions of the inhomogeneous porous medium equation with data in Lebesgue spaces, extending regularity results from the homogeneous case using geometric iteration and approximation techniques.

Contribution

It establishes sharp regularity results for inhomogeneous porous medium equations, providing explicit Hölder exponents depending on data integrability.

Findings

Solutions are locally Hölder continuous with an explicit exponent.

The regularity depends on the integrability of the inhomogeneous term.

The proof uses an approximation lemma and intrinsic geometric iteration.

Abstract

We show that locally bounded solutions of the inhomogeneous porous medium equation $$u_{t} - {\rm div} \left( m |u|^{m-1} \nabla u \right) = f \in L^{q,r}, \quad m >1 ,$$ are locally H\"older 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 H\"older exponent for solutions of the homogeneous case. The proof relies on an approximation lemma and geometric iteration in the appropriate intrinsic scaling.