Asymptotic expansions of solutions of fractional diffusion equations

TL;DR

This paper provides detailed asymptotic descriptions of solutions to linear and nonlinear fractional diffusion equations, extending previous methods to derive precise long-time behavior and expansions for these solutions.

Contribution

It introduces a new method to obtain asymptotic expansions for solutions of fractional diffusion equations, including nonlinear cases, building on and extending prior approaches.

Findings

Derived precise asymptotic behavior for linear fractional diffusion solutions.

Established a method for asymptotic expansions in nonlinear fractional diffusion.

Extended previous techniques to handle nonlinear terms in fractional PDEs.

Abstract

In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=0\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), \qquad u(x,0)=\varphi(x)\quad\mbox{in}\quad{\bf R}^N, $$ where $0<\theta<2$ and $\varphi\in L_K:=L^1({\bf R}^N,\,(1+|x|)^K\,dx)$ with $K\ge 0$. Furthermore, we develop the arguments in [15] and [18] and establish a method to obtain the asymptotic expansions of the solutions to a nonlinear fractional diffusion equation $$ \partial_t u+(-\Delta)^{\frac{\theta}{2}}u=|u|^{p-1}u\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), $$ where $0<\theta<2$ and $p>1+\theta/N$.