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 $${\partial }_t^{\alpha }u=-(-\Delta {)}^{\beta }u,\alpha \in (0,2),\beta \in (0,\infty ).$$ In this paper we provide the asymptotic behaviors and sharp upper bounds of $p(t,x)$ and its space and time fractional derivatives $$D_x^n(-{\Delta }_x{)}^{\gamma }{D}_t^{\sigma }{I}_t^{\delta }p(t,x),\forall n\in {\mathbb{Z}}_{+},\gamma \in [0,\beta ],\sigma ,\delta \in [0,\infty ),$$ 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.

References in corpus (2)

• Parabolic BMO estimates for pseudo-differential operators of arbitrary order
• An $L_q(L_p)$-theory for parabolic pseudo-differential equations: Calderón-Zygmund approach