Time fractional Poisson equations: Representations and estimates

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)

Abstract

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.

Original languageEnglish
Article number108311
JournalJournal of Functional Analysis
Volume278
Issue number2
DOIs
Publication statusPublished - 2020 Jan 15
Externally publishedYes

Keywords

  • Fundamental solution
  • Poisson equation
  • Subordinator

ASJC Scopus subject areas

  • Analysis

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