Time Fractional Poisson Equations: Representations and Estimates

TL;DR

This paper investigates solutions to time fractional Poisson equations, providing integral representations, fundamental solutions, and estimates under various conditions, advancing the mathematical understanding of these equations.

Contribution

It introduces integral representations and fundamental solutions for time fractional Poisson equations, including explicit estimates under specific subordinator conditions.

Findings

Existence and uniqueness of solutions established.

Integral kernel representation of solutions derived.

Two-sided estimates for fundamental solutions provided.

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 ${\cal L}$ 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 semigrou for ${\cal L}$ 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 homogenous 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.