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Gaussian estimates for fundamental solutions of higher-order parabolic equations with time-independent coefficients

Guoming Zhang

Abstract

We study the De Giorgi-Moser-Nash estimates of higher-order parabolic equations in divergence form with complex-valued, measurable, bounded, uniformly elliptic (in the sense of G$\mathring{a}$rding inequality) and time-independent coefficients. We also obtain Gaussian upper bounds and Hölder regularity estimates for the fundamental solutions of this class of parabolic equations.

Gaussian estimates for fundamental solutions of higher-order parabolic equations with time-independent coefficients

Abstract

We study the De Giorgi-Moser-Nash estimates of higher-order parabolic equations in divergence form with complex-valued, measurable, bounded, uniformly elliptic (in the sense of Grding inequality) and time-independent coefficients. We also obtain Gaussian upper bounds and Hölder regularity estimates for the fundamental solutions of this class of parabolic equations.

Paper Structure

This paper contains 5 sections, 10 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Technical lemmas
  4. The Hölder property from elliptic equations to prabolic equations
  5. Gaussian estimates for the fundamental solution of $\mathcal{H}$

Key Result

Lemma 3.1

Let $R>0$ and $u\in H^{m, 2}(Q_{R}(t, x))$ be a weak solution of $\mathcal{H}$ in $Q_{R}(t, x).$ Then for any $0<r<R,$ we have In particular, for $j=0,1, 2,...m-1,$

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • Lemma 3.4
  • Lemma 3.5
  • Lemma 3.6
  • Theorem 4.1
  • Remark 4.2
  • ...and 8 more