### 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)