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Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime

Karthik Adimurthi, Mitesh Modasiya

TL;DR

The paper addresses local Hölder regularity for bounded weak solutions to a nonlocal doubly nonlinear parabolic equation in a mixed singular-degenerate regime. It develops a partially unified intrinsic scaling approach combined with nonlocal De Giorgi iterations, tail controls, and energy estimates to achieve oscillation decay across near-zero and away-from-zero regimes. The authors construct a comprehensive framework of shrinking lemmas, propagation of measure, and energy inequalities to establish Hölder continuity under explicit range conditions on p, q, and s, while highlighting nonlocal features that cause instability as s approaches 0. The results extend known local and nonlocal regularity theories to a mixed regime, providing a rigorous, quantitative Hölder framework for a broad class of doubly nonlinear, nonlocal parabolic equations.

Abstract

We study local Hölder regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form \[ (|u|^{q-2}u)_t + \text{P.V.} \int_{\mathbb{R}^n} \frac{|u(x,t) - u(y,t)|^{p-2}(u(x,t)-u(y,t))}{|x-y|^{n+sp}} dy = 0, \] with $p\in (1,\infty)$, $q\in (1,\infty)$ and $s \in (0,1)$. Analogous Hölder continuity result in the local case is known in the purely singular case $\{1<p<2, p<q\}$, purely degenerate case $\{2<p, q<p\}$, scale invariant case $\{p=q\}$ and translation invariant case $\{q=2,1<p<\infty\}$. In the nonlocal setting, Hölder regularity is known when the equation is either translation invariant $\{q=2, 1<p<\infty\}$ or scale invariant $\{q=p, 1<p<\infty\}$ or purely degenerate case $\{2<p, q<p\}$. Similar strategy can be used to obtain Hölder regularity in the purely singular case $\{1<p<2, p<q\}$. In this paper, we adapt several ideas developed over the past few years and combine it with a new intrinsic scaling to prove Hölder regularity in the mixed singular-degenerate range $\max\{p,q,2\} < \min\left\{q + \tfrac{p-1}{1+\frac{n}{sp}}, 2 + \tfrac{p-1}{1+\frac{n}{sp}}\right\}$. The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at $s \rightarrow 0$. We note that the analogous regularity in the local problem remains open.

Hölder regularity of doubly nonlinear nonlocal quasilinear parabolic equations in some mixed singular-degenerate regime

TL;DR

The paper addresses local Hölder regularity for bounded weak solutions to a nonlocal doubly nonlinear parabolic equation in a mixed singular-degenerate regime. It develops a partially unified intrinsic scaling approach combined with nonlocal De Giorgi iterations, tail controls, and energy estimates to achieve oscillation decay across near-zero and away-from-zero regimes. The authors construct a comprehensive framework of shrinking lemmas, propagation of measure, and energy inequalities to establish Hölder continuity under explicit range conditions on p, q, and s, while highlighting nonlocal features that cause instability as s approaches 0. The results extend known local and nonlocal regularity theories to a mixed regime, providing a rigorous, quantitative Hölder framework for a broad class of doubly nonlinear, nonlocal parabolic equations.

Abstract

We study local Hölder regularity of bounded, weak solutions for the nonlocal quasilinear equations of the form with , and . Analogous Hölder continuity result in the local case is known in the purely singular case , purely degenerate case , scale invariant case and translation invariant case . In the nonlocal setting, Hölder regularity is known when the equation is either translation invariant or scale invariant or purely degenerate case . Similar strategy can be used to obtain Hölder regularity in the purely singular case . In this paper, we adapt several ideas developed over the past few years and combine it with a new intrinsic scaling to prove Hölder regularity in the mixed singular-degenerate range . The proof explicitly makes use of the nonlocal nature of the problem and as a consequence, our estimates are not stable at . We note that the analogous regularity in the local problem remains open.
Paper Structure (31 sections, 32 theorems, 215 equations, 5 figures)

This paper contains 31 sections, 32 theorems, 215 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Main theorems
  3. On historical development of intrinsic scaling
  4. A brief history of the problem
  5. Our approach to proving Hölder regularity to .
  6. Notations
  7. Function spaces
  8. Definitions
  9. Auxiliary Results
  10. Preliminary Estimates
  11. Shrinking Lemma
  12. Tail Estimates
  13. Partially unified proof of Hölder continuity when .
  14. Defining the constants
  15. de Giorgi iteration
  16. ...and 16 more sections

Key Result

Theorem 1.1

Let $p\in[2,\infty)$, $q =2$, $s\in(0,1)$ and let $u\in L^p(I;W^{s,p}_{\text{loc}}(\Omega))\cap L^\infty(I;L^q_{\text{loc}}(\Omega))\cap L^\infty(I;L^{p-1}_{sp}(\mathbb{R}^n))$ be any locally bounded, sign-changing weak solution to maineq. Then $u$ is locally Hölder continuous in $\Omega_T$, i.e., t for any $(x,t) \in B_{\frac{1}{2} R}(0)\times (-(\tfrac{1}{2} R)^{sp},0)$, the exponent $\mathfrak{

Figures (5)

  • Figure 1: Ranges of $p$ and $q$ for local doubly nonlinear equations
  • Figure 2: Ranges of $p$ and $q$ for nonlocal doubly nonlinear equations
  • Figure 3: Doubly singular ranges of $p$ and $q$
  • Figure 4: Mixed ranges of $p$ and $q$ from \ref{['holderparabolic']} For illustration purpose only for a fixed choice of $s$ and $n$, not to scale.
  • Figure 5: Defining the geometry

Theorems & Definitions (64)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Lemma 1.4
  • Definition 1.5
  • Theorem 1.6
  • Lemma 1.7
  • proof
  • Lemma 1.8
  • Lemma 1.9
  • ...and 54 more