Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Sharp asymptotic of solutions to some nonlocal parabolic equations

Agnid Banerjee, Abhishek Ghosh

Abstract

We show that if $u$ solves the fractional parabolic equation $(\partial_t - Δ)^s u = Vu$ in $B_5 \times (-25, 0]$ ($0<s<1$) such that $u(\cdot, 0) \not\equiv 0$, then the maximal vanishing order of $u$ in space-time at $(0,0)$ is upper bounded by $C\left(1+\|V\|_{C^{1}_{(x,t)}}^{1/2s}\right)$. As $s \to 1$, it converges to the sharp maximal order of vanishing due to Donnelly-Fefferman and Bakri. This quantifies a space like strong unique continuation result recently proved in [3]. The proof is achieved by means of a new quantitative Carleman estimate that we derive for the corresponding extension problem combined with a quantitative monotonicity in time result and a compactness argument.

Sharp asymptotic of solutions to some nonlocal parabolic equations

Abstract

We show that if solves the fractional parabolic equation in () such that , then the maximal vanishing order of in space-time at is upper bounded by . As , it converges to the sharp maximal order of vanishing due to Donnelly-Fefferman and Bakri. This quantifies a space like strong unique continuation result recently proved in [3]. The proof is achieved by means of a new quantitative Carleman estimate that we derive for the corresponding extension problem combined with a quantitative monotonicity in time result and a compactness argument.
Paper Structure (9 sections, 16 theorems, 159 equations, 1 figure)

This paper contains 9 sections, 16 theorems, 159 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Key ideas in the proof of Theorem \ref{['main']}:
  3. Preliminaries
  4. The main carleman estimate
  5. Quantitative doubling estimates
  6. Monotonicity in time
  7. Quantitative two-ball one-cylinder inequalities
  8. Vanishing estimate on cylinders
  9. Proof of Theorem \ref{['main']}

Key Result

Theorem 1.1

Fix $0<s<1.$ Let $u \in \operatorname{Dom}(H^s)$ solve in $B_5 \times (-25, 0]$. Assume that $u(\cdot, 0)\not\equiv 0$ in $B_1.$ Then there exists $\tilde{r}= \tilde{r}(u)>0$ such that for all $r \leq \tilde{r}(u)$ one has where $\mathcal{N}=M\left(\frac{1}{\int_{\mathbb B_1^+} U^2(X,0)x_{n+1}^adX}+ \operatorname{log}(M\Theta)+(\|V\|_1^{1/2s}+1)\right)$, $\Theta=\frac{\int_{\mathbb B_{5}^+\times

Figures (1)

  • Figure :

Theorems & Definitions (26)

  • Theorem 1.1: Quantitative space-like strong unique continuation
  • Lemma 2.1
  • Proposition 2.2
  • Lemma 2.3: EF_2003
  • proof
  • Lemma 2.4: Hardy type inequality
  • Lemma 2.5
  • Lemma 2.6: Trace inequality
  • Theorem 3.1
  • proof
  • ...and 16 more