Nonlocal Changing-Sign Perturbation Tempered Fractional Sub-Diffusion Model with Weak Singularity

Author:

Zhang Xinguang12ORCID,Chen Jingsong3,Chen Peng1,Li Lishuang1,Wu Yonghong2ORCID

Affiliation:

1. School of Mathematical and Informational Sciences, Yantai University, Yantai 264005, China

2. Department of Mathematics and Statistics, Curtin University, Perth, WA 6845, Australia

3. School of Statistics and Mathematics, Zhongnan University of Economics and Law, Wuhan 430073, China

Abstract

In this paper, we study the existence of positive solutions for a changing-sign perturbation tempered fractional model with weak singularity which arises from the sub-diffusion study of anomalous diffusion in Brownian motion. By two-step substitution, we first transform the higher-order sub-diffusion model to a lower-order mixed integro-differential sub-diffusion model, and then introduce a power factor to the non-negative Green function such that the linear integral operator has a positive infimum. This innovative technique is introduced for the first time in the literature and it is critical for controlling the influence of changing-sign perturbation. Finally, an a priori estimate and Schauder’s fixed point theorem are applied to show that the sub-diffusion model has at least one positive solution whether the perturbation is positive, negative or changing-sign, and also the main nonlinear term is allowed to have singularity for some space variables.

Funder

Natural Science Foundation of Shandong Province of China

ARC Discovery Project

Publisher

MDPI AG

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