On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms

Vo Van Au; Mokhtar Kirane; Nguyen Huy Tuan

doi:10.3934/dcdsb.2020174

Article Contents

2021, Volume 26, Issue 3: 1579-1613. Doi: 10.3934/dcdsb.2020174

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On a terminal value problem for a system of parabolic equations with nonlinear-nonlocal diffusion terms

1.
Institute of Fundamental and Applied Sciences, Duy Tan University, Ho Chi Minh City 700000, Vietnam
2.
Faculty of Natural Sciences, Duy Tan University, Da Nang, 550000, Vietnam
3.
LaSIE, Faculté des Sciences, Pole Sciences et Technologies, Université de La Rochelle, Avenue M. Crepeau, 17042 La Rochelle Cedex, France, NAAM Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia, RUDN University, 6 Miklukho-Maklay St, Moscow 117198, Russia
4.
Applied Analysis Research Group, Faculty of Mathematics, and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

^* Corresponding author: nguyenhuytuan@tdtu.edu.vn
^* Corresponding author: nguyenhuytuan@tdtu.edu.vn

Received: December 2019

Early access: May 2020

Published: March 2021

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Abstract

We study a terminal value parabolic system with nonlinear-nonlocal diffusions. Firstly, we consider the issue of existence and ill-posed property of a solution. Then we introduce two regularization methods to solve the system in which the diffusion coefficients are globally Lipschitz or locally Lipschitz under some a priori assumptions on the sought solutions. The existence, uniqueness and regularity of solutions of the regularized problem are obtained. Furthermore, The error estimates show that the approximate solution converges to the exact solution in $ L^2 $ norm and also in $ H^1 $ norm.

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Mathematics Subject Classification: 35B65, 35K05, 35K57, 35K99, 35Q92, 47H10.

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