Terminal value problem for nonlinear parabolic and pseudo-parabolic systems

Ho Duy Binh; Nguyen Van Tien; Vo Ngoc Minh; Nguyen Huu Can

doi:10.3934/dcdss.2023041

Article Contents

2023, Volume 16, Issue 10: 2839-2863. Doi: 10.3934/dcdss.2023041

This issue Previous Article Well-posedness results for nonlinear fractional diffusion equation with memory quantity Next Article Invariant measures for stochastic delay nonlocal lattice dynamical systems

Terminal value problem for nonlinear parabolic and pseudo-parabolic systems

1.
Division of Applied Mathematics, Science and Technology Advanced Institute, Van Lang University, Ho Chi Minh City, Vietnam
2.
Faculty of Applied Technology, School of Technology, Van Lang University, Ho Chi Minh City, Vietnam
3.
Faculty of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam
4.
Vietnam National University, Ho Chi Minh City, Vietnam
5.
Faculty of Math, FPT University HCM, Saigon Hi-Tech Park, Thu Duc City, Ho Chi Minh City, Vietnam
6.
Applied Analysis Research Group, Faculty of Mathematics and Statistics, Ton Duc Thang University, Ho Chi Minh City, Vietnam

^*Corresponding author: Nguyen Huu Can (nguyenhuucan@tdtu.edu.vn)
^*Corresponding author: Nguyen Huu Can (nguyenhuucan@tdtu.edu.vn)

Received: January 2023

Revised: February 2023

Early access: March 6, 2023

Published online: March 6, 2023

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Abstract

In this paper, we study on the backward problem for parabolic system and pseudo-parabolic system with conformable derivative. Two these problems have many applications in engineering such as image processing, geophysics, biology, etc. There are two main contributions in this paper. The first major result deals with ill-posedness and regularization for terminal value problem for parabolic system. By applying a new truncation method, we construct a regularized solution. We investigate the existence and uniqueness of regularized problem. Under some suitable conditions of the terminal data, we provide the error estimate between the mild and regularized solutions. The second contribution concerns the existence of a global solution of pseudo-parabolic system. Finally, we also obtain the convergence of the mild solution as the order of derivative approaches $ 1^-. $

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Mathematics Subject Classification: 35R11, 35B65, 26A33.

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