# Time fractional Poisson equations: Representations and estimates

Zhen Qing Chen*, Panki Kim, Takashi Kumagai, Jian Wang

*この研究の対応する著者

8 被引用数 (Scopus)

## 抄録

In this paper, we study existence and uniqueness of strong as well as weak solutions for general time fractional Poisson equations. We show that there is an integral representation of the solutions of time fractional Poisson equations with zero initial values in terms of semigroup for the infinitesimal spatial generator L and the corresponding subordinator associated with the time fractional derivative. This integral representation has an integral kernel q(t,x,y), which we call the fundamental solution for the time fractional Poisson equation, if the semigroup for L has an integral kernel. We further show that q(t,x,y) can be expressed as a time fractional derivative of the fundamental solution for the homogeneous time fractional equation under the assumption that the associated subordinator admits a conjugate subordinator. Moreover, when the Laplace exponent of the associated subordinator satisfies the weak scaling property and its distribution is self-decomposable, we establish two-sided estimates for the fundamental solution q(t,x,y) through explicit estimates of transition density functions of subordinators.

本文言語 English 108311 Journal of Functional Analysis 278 2 https://doi.org/10.1016/j.jfa.2019.108311 Published - 2020 1月 15 はい

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