On the H\"older regularity of signed solutions to a doubly nonlinear equation. Part III

TL;DR

This paper proves that solutions to a class of doubly nonlinear parabolic equations, which can change sign, are locally H"older continuous, using intrinsic scaling and positivity expansion techniques.

Contribution

It establishes the local H"older regularity for sign-changing solutions to a doubly nonlinear parabolic equation, extending previous regularity results to a broader class of solutions.

Findings

Solutions are locally H"older continuous.

The proof uses space expansion of positivity.

Intrinsic scaling balances the double singularity.

Abstract

We establish the local H\"older continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is \[ \partial_t\big(|u|^{q-1}u\big)-\Delta_p u=0,\quad 1<p<2,\quad 0<p-1<q. \] The proof exploits the space expansion of positivity for the singular, parabolic $p$-Laplacian and employs the method of intrinsic scaling by carefully balancing the double singularity.