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

Renhai Wang (), Hoang van Dai, Nguyen Anh Tuan and Nguyen Huu Can
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Renhai Wang: School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, P. R. China
Hoang van Dai: Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam3Vietnam National University, Ho Chi Minh City, Vietnam
Nguyen Anh Tuan: Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam5Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam
Nguyen Huu Can: Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

FRACTALS (fractals), 2023, vol. 31, issue 10, 1-13

Abstract: In this paper, we are interested in studying the fractional diffusion equation with Riemann–Liouville as follows: D0+αy − y xx = 0, 0 < x < π, 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 Lp 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: Fractional Diffusion Equation; Riemann–Liouville; Regularity; Nonlocal in Time Condition (search for similar items in EconPapers)
Date: 2023
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DOI: 10.1142/S0218348X2340193X

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