Asymptotic behaviors of fundamental solution and its derivatives related to space-time fractional differential equations

Abstract

Let $p(t,x)$ be the fundamental solution to the problem $$ \partial_{t}^{\alpha}u=-(-\Delta)^{\beta}u, \quad \alpha\in (0,2), \, \beta\in (0,\infty). $$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$ D_{x}^{n}(-\Delta_x)^{\gamma}D_{t}^{\sigma}I_{t}^{\delta}p(t,x), \quad \forall\,\, n\in\mathbb{Z}_{+}, \,\, \gamma\in[0,\beta],\,\, \sigma, \delta \in[0,\infty), $$ 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.