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Asymptotic expansions of solutions of fractional diffusion equations (1610.09789v1)
Published 31 Oct 2016 in math.AP
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 $\varphi\in L_K:=L1({\bf R}N,\,(1+|x|)K\,dx)$ 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 $p>1+\theta/N$.