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Upper bounds of nodal sets for Gevrey regular parabolic equations

Guher Camliyurt, Igor Kukavica, Linfeng Li

Abstract

We consider the size of the nodal set of the solution of the second order parabolic-type equation with Gevrey regular coefficients. We provide an upper bound as a function of time. The dependence agrees with a sharp upper bound when the coefficients are analytic.

Upper bounds of nodal sets for Gevrey regular parabolic equations

Abstract

We consider the size of the nodal set of the solution of the second order parabolic-type equation with Gevrey regular coefficients. We provide an upper bound as a function of time. The dependence agrees with a sharp upper bound when the coefficients are analytic.
Paper Structure (3 sections, 5 theorems, 62 equations)

This paper contains 3 sections, 5 theorems, 62 equations.

Table of Contents

  1. Introduction
  2. G·evrey regular parabolic problem and the main result
  3. Proof of the main theorem

Key Result

Theorem 2.1

Suppose that $u(x,t)$ is the solution to para01 and para0 on $\mathbb T^d\times I$ with $u_0 \in H^1 (\mathbb T^d)$ not identically zero. Assume that $v$ and $w$ satisfy para03--para02 and EQ52--EQ72. Then, for each $t\in I$, we have where $C>0$ is a constant depending on $M_0$, $M_1$, $K_v$ and $K_w$.

Theorems & Definitions (9)

  • Theorem 2.1
  • Lemma 3.1: K2
  • Lemma 3.2: K2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • proof : Proof of Lemma \ref{['L03']}
  • Remark 3.5
  • proof : Proof of Theorem \ref{['T01']}