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

Let $p(t,x)$ be the fundamental solution to the problem $${\mathrm{\partial }}_t^{\alpha }u=-(-\mathrm{\Delta }{)}^{\beta }u,\alpha \in (0,2),\beta \in (0,\mathrm{\infty })\mathrm{.}$$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$D_x^n(-{\mathrm{\Delta }}_x{)}^{\gamma }{D}_t^{\sigma }{I}_t^{\delta }p(t,x),\mathrm{\forall }n\in {\mathbb{Z}}_{+},\gamma \in [0,\beta ],\sigma ,\delta \in [0,\mathrm{\infty }),$$ where $D_x^n$ is a partial derivative of order $n$ with respect to $x$, $(-{\mathrm{\Delta }}_x{)}^{\gamma }$ is a fractional Laplace operator and $D_t^{\sigma }$ and $I_t^{\delta }$ are Riemann-Liouville fractional derivative and integral respectively.