TY - JOUR

T1 - Time fractional Poisson equations

T2 - Representations and estimates

AU - Chen, Zhen Qing

AU - Kim, Panki

AU - Kumagai, Takashi

AU - Wang, Jian

N1 - Funding Information:
The research of Zhen-Qing Chen is partially supported by Simons Foundation Grant 520542 and a Victor Klee Faculty Fellowship at the University of Washington. The research of Panki Kim is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1E1A1A01941893). The research of Takashi Kumagai is supported by JSPS KAKENHI Grant Number JP17H01093 and by the Alexander von Humboldt Foundation. The research of Jian Wang is supported by the National Natural Science Foundation of China (Nos. 11522106 and 11831014), the Fok Ying Tung Education Foundation (No. 151002), the Program for Probability and Statistics: Theory and Application (No. IRTL1704) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ). The main results of this paper, including the two-sided estimates on the fundamental solution q(t,x,y) for the time fractional Poisson equations, have been reported in several conferences including Fractional PDEs: theory, algorithms and applications held from June 18 to 22, 2018 at Brown University, Providence, The Fifth IMS Asia Pacific Rim Meeting from June 26 to 29, 2018 in Singapore, Workshop on Dynamics, Control and Numerics for Fractional PDEs from December 5 to 7, 2018 at San Juan, Puerto Rico, and Non Standard Diffusions in Fluids, Kinetic Equations and Probability from December 10 to 14, 2018 at CIRM, Marseille. We thank the organizers for the invitations and participants for their interest.
Funding Information:
The research of Zhen-Qing Chen is partially supported by Simons Foundation Grant 520542 and a Victor Klee Faculty Fellowship at the University of Washington . The research of Panki Kim is supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIP) (No. 2016R1E1A1A01941893 ). The research of Takashi Kumagai is supported by JSPS KAKENHI Grant Number JP17H01093 and by the Alexander von Humboldt Foundation . The research of Jian Wang is supported by the National Natural Science Foundation of China (Nos. 11522106 and 11831014 ), the Fok Ying Tung Education Foundation (No. 151002 ), the Program for Probability and Statistics: Theory and Application (No. IRTL1704 ) and the Program for Innovative Research Team in Science and Technology in Fujian Province University (IRTSTFJ).
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2020/1/15

Y1 - 2020/1/15

N2 - 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.

AB - 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.

KW - Fundamental solution

KW - Poisson equation

KW - Subordinator

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U2 - 10.1016/j.jfa.2019.108311

DO - 10.1016/j.jfa.2019.108311

M3 - Article

AN - SCOPUS:85072831194

SN - 0022-1236

VL - 278

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 2

M1 - 108311

ER -