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Continuity estimates for degenerate parabolic double-phase equations via nonlinear potentials

Qifan Li

TL;DR

The paper develops a regularity theory for bounded weak solutions of degenerate parabolic double-phase equations by linking nonlinear potential theory to a De Giorgi–Kilpeläinen–Malý framework in intrinsic cylinders. It proves local continuity with explicit oscillation decay in terms of elliptic Riesz potentials $F_p$ and $G_p$, under nonlinear Kato-type conditions $f\in K_p$ and $g\in \tilde K_p$, and structural assumptions on the coefficient $a(x,t)$. The method combines Caccioppoli inequalities, logarithmic estimates, and two complementary De Giorgi alternatives (first and second) to derive decay of oscillation across shrinking intrinsic cylinders, yielding Hölder-type continuity and quantitative bounds. The results extend to the parabolic $p$- and $(p,q)$-phase settings and provide a potential-theoretic characterization of regularity via $F_p$ and $G_p$, with uniform estimates in the $p$-phase and refined bounds when $a_0>0$, offering a robust regularity framework for parabolic double-phase problems.

Abstract

In this article, we study regularity properties for degenerate parabolic double-phase equations. We establish continuity estimates for bounded weak solutions in terms of elliptic Riesz potentials on the right-hand side of the equation.

Continuity estimates for degenerate parabolic double-phase equations via nonlinear potentials

TL;DR

The paper develops a regularity theory for bounded weak solutions of degenerate parabolic double-phase equations by linking nonlinear potential theory to a De Giorgi–Kilpeläinen–Malý framework in intrinsic cylinders. It proves local continuity with explicit oscillation decay in terms of elliptic Riesz potentials and , under nonlinear Kato-type conditions and , and structural assumptions on the coefficient . The method combines Caccioppoli inequalities, logarithmic estimates, and two complementary De Giorgi alternatives (first and second) to derive decay of oscillation across shrinking intrinsic cylinders, yielding Hölder-type continuity and quantitative bounds. The results extend to the parabolic - and -phase settings and provide a potential-theoretic characterization of regularity via and , with uniform estimates in the -phase and refined bounds when , offering a robust regularity framework for parabolic double-phase problems.

Abstract

In this article, we study regularity properties for degenerate parabolic double-phase equations. We establish continuity estimates for bounded weak solutions in terms of elliptic Riesz potentials on the right-hand side of the equation.

Paper Structure

This paper contains 6 sections, 15 theorems, 434 equations.

Table of Contents

  1. Introduction
  2. Statement of the main result
  3. Caccioppoli inequalities
  4. The first alternative
  5. The second alternative
  6. Proof of the main result

Key Result

Theorem 2.4

Let $u$ be a locally bounded weak solution to the parabolic double-phase equation parabolic in the sense of Definition weak solution, where the vector field $A$ fulfills the structure conditions A. Assume that $2<p<q\leq p+\alpha$, $2<p<n$, $f\in K_p$ and $g\in \tilde{K}_p$. Then, $u$ is locally con holds for any $r<\delta_1R_0$. Here, the constants $b_0$, $\alpha_1$, $\alpha_2$, $\gamma$ and $c$

Theorems & Definitions (29)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • Lemma 3.4
  • ...and 19 more