Asymptotic behavior of solutions to a space fractional diffusion equation

TL;DR

This paper analyzes the long-term behavior of solutions to a one-dimensional space fractional diffusion equation with Caputo derivatives, providing improved decay estimates and convergence rates depending on boundary conditions.

Contribution

It offers new decay estimates and convergence rates for solutions on the half-line, considering different boundary conditions, which advances understanding of fractional diffusion dynamics.

Findings

Solutions converge in L^p to self-similar solutions or decay to zero

Convergence rates are explicitly provided

Behavior depends on boundary conditions

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.