Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Asymptotic behavior of solutions to a space fractional diffusion equation

Barbara Łupińska, Piotr Rybka

TL;DR

This work analyzes the long-time behavior of solutions to a space-fractional diffusion equation with Caputo derivative on the half-line under Dirichlet or Neumann boundary conditions at $x=0$. It develops a self-similar fundamental solution ${\mathscr{E}}_t$ built from a profile $\Phi$ satisfying a fractional ODE and derives sharp decay bounds for $\Phi$ and its derivative, enabling convolution-based solution representations. By decomposing solutions into kernels $w_1$ and $w_2$ and applying Marcinkiewicz interpolation, the paper proves precise $L^p$ decay rates and convergence results: Dirichlet data lead to extinguishing behavior, while Neumann data yield convergence to a multiple of ${\mathscr{E}}_t$, i.e., to $2m{\mathscr{E}}_t$ with a rate $t^{- rac{2-1/p}{1+\alpha}}$ for $1<p<\infty$. The results rely on boundedness and decay properties of the fundamental solution, and hinge on compactly supported initial data to establish the key estimates.

Abstract

We improve the time decay estimates of solutions to the one-dimensional fractional diffusion equation involving the Caputo derivative. The equation is considered on the half-line. Depending on the boundary condition, we show that solutions converge in $L^p$, $p>1$ to a multiple of the self-similar solutions or decay to zero. The convergence rate is provided.

Asymptotic behavior of solutions to a space fractional diffusion equation

TL;DR

This work analyzes the long-time behavior of solutions to a space-fractional diffusion equation with Caputo derivative on the half-line under Dirichlet or Neumann boundary conditions at . It develops a self-similar fundamental solution built from a profile satisfying a fractional ODE and derives sharp decay bounds for and its derivative, enabling convolution-based solution representations. By decomposing solutions into kernels and and applying Marcinkiewicz interpolation, the paper proves precise decay rates and convergence results: Dirichlet data lead to extinguishing behavior, while Neumann data yield convergence to a multiple of , i.e., to with a rate for . The results rely on boundedness and decay properties of the fundamental solution, and hinge on compactly supported initial data to establish the key estimates.

Abstract

We improve the time decay estimates of solutions to the one-dimensional fractional diffusion equation involving the Caputo derivative. The equation is considered on the half-line. Depending on the boundary condition, we show that solutions converge in , to a multiple of the self-similar solutions or decay to zero. The convergence rate is provided.

Paper Structure

This paper contains 6 sections, 9 theorems, 74 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Fractional operators
  4. The fundamental solution
  5. The decay properties of the profile function $\Phi$
  6. The convergence result

Key Result

Proposition 1

Let us suppose ${\mathscr{E}}$ is the fundamental solution defined above. Then: (1) ${\mathscr{E}}\in C^2(0,\infty)^2$; (2) ${\mathscr{E}}$ is positive for all $x,\ t>0$; (3) for all $t>0$ function ${\mathscr{E}}(\cdot,t)$ is decreasing; (4) for all $t>0$ function ${\mathscr{E}}(\cdot,t)$ is in $L^1

Theorems & Definitions (15)

  • Remark 1: see Kilbas
  • Proposition 1
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Remark 2
  • Proposition 2
  • ...and 5 more