Sharp and improved regularity for a class of doubly degenerate parabolic Pdes

TL;DR

This paper establishes sharp local H"older regularity estimates for solutions of a class of doubly degenerate parabolic PDEs using intrinsic scaling and geometric methods, improving previous results.

Contribution

It introduces a unified approach to derive optimal regularity estimates for nonlinear degenerate evolution PDEs, including Liouville type results and explicit examples.

Findings

Sharp H"older regularity estimates obtained

Liouville type results for entire solutions established

Explicit examples illustrating the applicability provided

Abstract

In this manuscript we establish local H\"older regularity estimates for bounded solutions of a certain class of doubly degenerate evolution PDEs. By making use of intrinsic scaling techniques and geometric tangential methods, we derive sharp regularity estimates for such models, which depend only on universal and compatibility parameters of the problem. In such a scenario, our results are natural improvements for former ones in the context of nonlinear evolution PDEs with degenerate structure via a unified approach. As a consequence for our findings and approach, we address a Liouville type result for entire solutions of a related homogeneous problem with frozen coefficients and asymptotic estimates under a certain approximating regime, which may have their own mathematical interest. We also deliver explicit examples of degenerate PDEs where our results take place.