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Pointwise regularity of solutions for fully fractional parabolic equations

Yahong Guo, Qizhen Shen, Jiongduo Xie

Abstract

This paper investigates the higher pointwise regularity of nonnegative classical solutions for fully fractional parabolic equations $(\partial_t -Δ)^{s} u = f,$ where $s\in(0,1)$. We establish $C^{k+α+2s}$ or $C^{k+α+2s,\ln} (k\geq 0,α\in[0,1))$ pointwise regularity according to $α+2s\notin \mathbb{Z}$ or $α+2s\in \mathbb{Z}$, which imply the classical local regularity directly. We provide a simplified and unified proof by introducing novel equivalent definitions for pointwise function spaces. Moreover, the equivalent integral representation and directional average for fractional heat kernel play an important role in our discussion.

Pointwise regularity of solutions for fully fractional parabolic equations

Abstract

This paper investigates the higher pointwise regularity of nonnegative classical solutions for fully fractional parabolic equations where . We establish or pointwise regularity according to or , which imply the classical local regularity directly. We provide a simplified and unified proof by introducing novel equivalent definitions for pointwise function spaces. Moreover, the equivalent integral representation and directional average for fractional heat kernel play an important role in our discussion.
Paper Structure (11 sections, 21 theorems, 151 equations, 1 figure)

This paper contains 11 sections, 21 theorems, 151 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Notation and definitions
  3. Main Results
  4. Main Ideas of the Proof
  5. Estimate of the External Part $v_{r}$
  6. Estimate of the Internal Part $w_{1}$
  7. Estimate for $S_{r}$
  8. Estimate for $u_{P}$
  9. Estimate for $T_{r}$ with $2s+\alpha\notin\mathbb{Z}$
  10. Estimate for $T_{r}$ with $2s+\alpha\in\mathbb{Z}$
  11. Appendix: Equivalent Characterizations of Pointwise Function Spaces

Key Result

Theorem 1

Assume that $f$ is a nonnegative function in $\mathbb{R}^{n}\times\mathbb{R}$ and $u$ is a nonnegative solution of (1.0). Then there exists a constant $C=C(n,k,\alpha,s)>0$ such that for each $(x_0,t_0) \in \mathbb{R}^n \times \mathbb{R}$, if $f\in C_{1}^{k+\alpha}(x_{0},t_{0};1)$, then

Figures (1)

  • Figure 1: Partition for the solution

Theorems & Definitions (43)

  • Definition 1.1: Parabolic polynomial Lianyy
  • Definition 1.2
  • Remark 1
  • Remark 2
  • Definition 1.3
  • Theorem 1
  • Corollary 1
  • Remark 3
  • Corollary 2
  • Corollary 3
  • ...and 33 more