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WELL-POSEDNESS AND REGULARIZATION FOR NONLOCAL DIFFUSION EQUATION WITH RIEMANN–LIOUVILLE DERIVATIVE

RENHAI WANG

https://orcid.org/0000-0002-6488-2474

School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, P. R. China

E-mail Address: rwang-math@outlook.com

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HOANG VAN DAI

https://orcid.org/0009-0006-4431-0993

Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam

Vietnam National University, Ho Chi Minh City, Vietnam

E-mail Address: hoangdai.bk@gmail.com

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NGUYEN ANH TUAN

https://orcid.org/0000-0002-8757-9742

Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam

Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam

E-mail Address: nguyenanhtuan@vlu.edu.vn

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, and

NGUYEN HUU CAN

https://orcid.org/0000-0001-6198-1015

Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

E-mail Address: nguyenhuucan@tdtu.edu.vn

Corresponding author. Document Type: Research Article.

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https://doi.org/10.1142/S0218348X2340193XCited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue on Fractal AI-Based Analyses and Applications to Complex Systems IV
Guest Editors: Yeliz Karaca, Dumitru Baleanu, Majaz Moonis, Yu-Dong Zhang and Osvaldo Gervasi

Abstract

In this paper, we are interested in studying the fractional diffusion equation with Riemann–Liouville as follows:

D α 0 + y - y x x = 0, 0 < x < π, <math display="block" altimg="eq-00001.gif"><mrow><msubsup><mrow><mstyle mathvariant="bold"><mi>D</mi></mstyle></mrow><mrow><msup><mrow><mn>0</mn></mrow><mrow><mo stretchy="false" class="MathClass-bin">+</mo></mrow></msup></mrow><mrow><mspace width="0.02cm" class="hspace"></mspace><mi>α</mi></mrow></msubsup><mi>y</mi><mo stretchy="false" class="MathClass-bin">-</mo><msub><mrow><mi>y</mi></mrow><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo class="MathClass-rel">=</mo><mn>0</mn><mo class="MathClass-punc">,</mo><mspace width="1em" class="quad"></mspace><mn>0</mn><mo class="MathClass-rel">&lt;</mo><mi>x</mi><mo class="MathClass-rel">&lt;</mo><mi>π</mi><mo class="MathClass-punc">,</mo></mrow></math>

with nonlocal in time condition. We are going to study the well-posedness of the above problem with some assumptions of the input data. On the other hand, in Hilbert scale and

$L^{p}$ spaces, we provide several estimates of regularity results of the mild solution. We also establish the evaluation for gradient term of the mild solution. We also show that the nonlocal problem is ill-posed in the sense of Hadamard. We also derive the regularity result by applying Fourier truncation method. The main tool of the paper is to use some estimates of Wright functions and Sobolev embeddings. In addition, we also obtain a lower bound of the solution according to the input data.

Keywords:

References

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Vol. 31, No. 10

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History

Received 15 June 2023

Accepted 18 October 2023

Published: December 9, 2023

Information

This is an Open Access article in the “Special Issue on Fractal AI-Based Analyses and Applications to Complex Systems: Part IV”, edited by Yeliz Karaca https://orcid.org/0000-0001-8725-6719 (University of Massachusetts Chan Medical School, USA), Dumitru Baleanu https://orcid.org/0000-0002-0286-7244 (Cankaya University, Turkey & Institute of Space Sciences, Romania), Majaz Moonis https://orcid.org/0000-0002-9747-0003 (University of Massachusetts Chan Medical School, USA), Yu-Dong Zhang https://orcid.org/0000-0002-4870-1493 (University of Leicester, Leicester, UK) & Osvaldo Gervasi https://orcid.org/0000-0003-4327-520X (Perugia University, Perugia, Italy) published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution-NonCommercial 4.0 (CC BY-NC) License which permits use, distribution and reproduction in any medium, provided that the original work is properly cited and is used for non-commercial purposes.

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WELL-POSEDNESS AND REGULARIZATION FOR NONLOCAL DIFFUSION EQUATION WITH RIEMANN–LIOUVILLE DERIVATIVE

Abstract

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