On the Hölder continuity of signed solutions to doubly nonlinear parabolic equations in the mixed degenerate/singular cases

TL;DR

The paper proves local Hölder continuity for sign-changing weak solutions to the doubly nonlinear parabolic equation $\partial_t(|u|^{q-1}u) - \nabla\cdot(|Du|^{p-2}Du)=0$ under structural conditions on the flux $\mathbf{A}$ and two complementary parameter regimes: $0<q<p-1$ with $p<2$, or $q>p-1$ with $p>2$.A central novelty is the development of integral Harnack-type inequalities for sign-changing solutions, which, together with De Giorgi-type lemmas and energy estimates, yield robust oscillation decay in parabolic cylinders $Q_{r,b\omega^{q-p+1}r^p}$.The authors also establish expansion-of-positivity results that enable propagation of lower bounds (and similar upper bounds) in time, allowing the Hölder regularity argument to cover degenerate/singular interfaces and sign-changing data.The results extend known Hölder regularity to previously unhandled degenerate/singular regimes and to sign-changing solutions, providing a framework that unifies porous medium, parabolic $p$-Laplacian, and Trudinger-type dynamics with measurable coefficients.

Abstract

We prove the Hölder continuity of sign-changing solutions to the equation of the type $$\frac{\partial}{\partial t}\big(|u|^{q-1} u\big)- div\Big(|D u|^{p-2}\,D u\Big)=0,$$ where numbers $p$, $q$ satisfy the conditions $$0<q<p-1\quad \text{and}\quad p<2,$$ or $$q>p-1\quad\text{and}\quad p>2.$$ Our proof uses new versions of the integral Harnack type inequalities for sign-changing solutions.

Theorems & Definitions (24)

• Theorem 1.1
• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Theorem 1.4
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4