Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations

摘要

Let $p(t,x)$ be the fundamental solution to the problem $$ partial_{t}^{lpha}u=-(-Delta)^{eta}u, quad lphain (0,2), , etain (0,infty). $$ If $lpha,etain (0,1)$, then the kernel $p(t,x)$ becomes the transition density of a L'evy process delayed by an inverse subordinator. In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$ D_{x}^{n}(-Delta_x)^{gamma}D_{t}^{sigma}I_{t}^{delta}p(t,x), quad orall,, ninmathbb{Z}_{+}, ,, gammain[0,eta],,, sigma, delta in[0,infty), $$ where $D_{x}^n$ is a partial derivative of order $n$ with respect to $x$, $(-Delta_x)^{gamma}$ is a fractional Laplace operator and $D_{t}^{sigma}$ and $I_{t}^{delta}$ are Riemann-Liouville fractional derivative and integral respectively.