Asymptotic expansions of solutions of fractional diffusion equations

Abstract

In this paper we obtain the precise description of the asymptotic behavior of the solution $u$ of $$ \partial_t u+(-Δ)^{\fracθ{2}}u=0\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), \qquad u(x,0)=φ(x)\quad\mbox{in}\quad{\bf R}^N, $$ where $0<\theta <2$ and $\phi \in L_K:=L^1({\mathbf{R}}^N,(1+|x|{)}^{K}dx)$ 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 $$ \partial_t u+(-Δ)^{\fracθ{2}}u=|u|^{p-1}u\quad\mbox{in}\quad{\bf R}^N\times(0,\infty), $$ where $0<\theta <2$ and $p>1+\theta /N$.

References in corpus (1)

• Asymptotics for a nonlinear integral equation with a generalized heat kernel

Cited by in corpus (3)

• Spatial-decay of solutions to the quasi-geostrophic equation with the critical and the super-critical dissipation
• Time evolution of the Navier-Stokes flow in far-field
• Parabolic-scalings on large-time behavior of the incompressible Navier--Stokes flow