## Abstract

In this paper we obtain the precise description of the asymptotic behavior of the solution u of the fractional diffusion equation ∂_{t}u + (-Δ)^{θ/2} u = 0 in R^{N} × (0,∞) with the initial data φ ∈ L_{K} := L^{1}(R^{N}, (1 + |x|)^{K} dx), where 0 < θ < 2 and K ≥ 0. This enables us to obtain the asymptotic behavior of the hot spots of the solution u. Furthermore, we develop the arguments in [K. Ishige and T. Kawakami, Math. Ann., 353 (2012), pp. 161-192] and [K. Ishige, T. Kawakami, and K. Kobayashi, J. Evol. Equ., 14 (2014), pp. 749-777] and establish a method to obtain the asymptotic expansions of the solutions to inhomogeneous fractional diffusion equations and nonlinear fractional diffusion equations.

Original language | English |
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Pages (from-to) | 2167-2190 |

Number of pages | 24 |

Journal | SIAM Journal on Mathematical Analysis |

Volume | 49 |

Issue number | 3 |

DOIs | |

Publication status | Published - 2017 Jan 1 |

## Keywords

- Anomalous diffusion
- Asymptotic expansion
- Fractional diffusion equation
- Hot spot
- Semilinear parabolic equation

## ASJC Scopus subject areas

- Analysis
- Computational Mathematics
- Applied Mathematics