### Abstract

Let p(t; x) be the fundamental solution to the problem (Formula Presented), then the kernel p(t; x) becomes the transition density of a Lévy 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 (Formula Presented); where D^{n} _{x} is a partial derivative of order n with respect to x, (-Δ_{x})^{γ} is a fractional Laplace operator and D^{σ} _{t} and I^{σ} _{t} are Riemann-Liouville fractional derivative and integral respectively.

Original language | English |
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Pages (from-to) | 929-967 |

Number of pages | 39 |

Journal | Journal of the Korean Mathematical Society |

Volume | 53 |

Issue number | 4 |

DOIs | |

Publication status | Published - 2016 |

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### Keywords

- Asymptotic behavior
- Fractional diffusion
- Fundamental solution
- Lévy process
- Space-time fractional differential equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

**Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations.** / Kim, Kyeong Hun; Lim, Sungbin.

Research output: Contribution to journal › Article

*Journal of the Korean Mathematical Society*, vol. 53, no. 4, pp. 929-967. https://doi.org/10.4134/JKMS.j150343

}

TY - JOUR

T1 - Asymptotic behaviors of fundamental solution and its derivatives to fractional diffusion-wave equations

AU - Kim, Kyeong Hun

AU - Lim, Sungbin

PY - 2016

Y1 - 2016

N2 - Let p(t; x) be the fundamental solution to the problem (Formula Presented), then the kernel p(t; x) becomes the transition density of a Lévy 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 (Formula Presented); where Dn x is a partial derivative of order n with respect to x, (-Δx)γ is a fractional Laplace operator and Dσ t and Iσ t are Riemann-Liouville fractional derivative and integral respectively.

AB - Let p(t; x) be the fundamental solution to the problem (Formula Presented), then the kernel p(t; x) becomes the transition density of a Lévy 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 (Formula Presented); where Dn x is a partial derivative of order n with respect to x, (-Δx)γ is a fractional Laplace operator and Dσ t and Iσ t are Riemann-Liouville fractional derivative and integral respectively.

KW - Asymptotic behavior

KW - Fractional diffusion

KW - Fundamental solution

KW - Lévy process

KW - Space-time fractional differential equation

UR - http://www.scopus.com/inward/record.url?scp=84975292473&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84975292473&partnerID=8YFLogxK

U2 - 10.4134/JKMS.j150343

DO - 10.4134/JKMS.j150343

M3 - Article

AN - SCOPUS:84975292473

VL - 53

SP - 929

EP - 967

JO - Journal of the Korean Mathematical Society

JF - Journal of the Korean Mathematical Society

SN - 0304-9914

IS - 4

ER -