Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 2016; 53(4): 929-967

Printed July 1, 2016

https://doi.org/10.4134/JKMS.j150343

Copyright © The Korean Mathematical Society.

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

Kyeong-Hun Kim and Sungbin Lim

Korea University, Korea University

Abstract

Let $p(t,x)$ be the fundamental solution to the problem $$ \partial_{t}^{\alpha}u=-(-\Delta)^{\beta}u, \quad \alpha\in (0,2), \, \beta\in (0,\infty). $$ If $\alpha,\beta\in (0,1)$, 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 $$ D_{x}^{n}(-\Delta_x)^{\gamma}D_{t}^{\sigma}I_{t}^{\delta}p(t,x), \quad \forall\,\, n\in\mathbb{Z}_{+}, \,\, \gamma\in[0,\beta],\,\, \sigma, \delta \in[0,\infty), $$ 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.

Keywords: fractional diffusion, L\'{e}vy process, asymptotic behavior, fundamental solution, space-time fractional differential equation

MSC numbers: 60J60, 60K99, 26A33, 35A08, 35R11, 45K05, 45M05