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Large time behaviour of the fractional heat equation associated with the Dunkl Laplacian

Suman Mukherjee

Abstract

We consider the fractional heat equation associated with the Dunkl Laplacian and prove that the weak solutions to this equation converge to the fundamental solution as time becomes large, provided the initial data is an integrable function with respect to the associated measure. As an application, we also prove a similar result for the corresponding nonlinear equation.

Large time behaviour of the fractional heat equation associated with the Dunkl Laplacian

Abstract

We consider the fractional heat equation associated with the Dunkl Laplacian and prove that the weak solutions to this equation converge to the fundamental solution as time becomes large, provided the initial data is an integrable function with respect to the associated measure. As an application, we also prove a similar result for the corresponding nonlinear equation.
Paper Structure (10 sections, 5 theorems, 56 equations)

This paper contains 10 sections, 5 theorems, 56 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and notations
  3. The Dunkl setup
  4. Dunkl operators, Dunkl kernel and Dunkl transform
  5. Dunkl translation and Dunkl convolution
  6. Dunkl heat kernel and Fractional Dunkl heat kernel
  7. Proof of Theorem \ref{['main thm linear part']}
  8. Proof for the case $\alpha=1$
  9. Proof for the case $0<\alpha <1$
  10. Proof of Theorem \ref{['main thm non-linear part']}

Key Result

Theorem 1.1

Let $1\leq p\leq \infty$ and $u_0 \in L^1(d\mu_k)$ be such that Then

Theorems & Definitions (10)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.3
  • Theorem 2.1
  • Theorem 3.1
  • Proposition 4.1
  • proof
  • proof : Proof of Theorem \ref{['main thm linear part']} for $\alpha=1$
  • proof : Proof of Theorem \ref{['main thm linear part']} for $0<\alpha < 1$
  • proof : Proof of Theorem \ref{['main thm non-linear part']}