On a final value problem for a nonlinear fractional pseudo-parabolic equation

Vo Van Au; Hossein Jafari; Zakia Hammouch; Nguyen Huy Tuan; Vo Van Au; Hossein Jafari; Zakia Hammouch; Nguyen Huy Tuan

doi:10.3934/era.2020088

Electronic Research Archive

2021, Volume 29, Issue 1: 1709-1734. doi: 10.3934/era.2020088

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On a final value problem for a nonlinear fractional pseudo-parabolic equation

1.
Division of Applied Mathematics, Thu Dau Mot University, Thu Dau Mot City, Vietnam
2.
Department of Mathematics, University of Mazandaran, Babolsar, Iran, Department of Mathematical Sciences, University of South Africa, UNISA0003, South Africa
3.
Department of Medical Research, China Medical University Hospital, China Medical University, Taichung 110122, Taiwan
4.
Department of Mathematics, FSTE Moulay Ismail University of Meknes, BP 509 Boutalamine, Errachidia 52000, Morocco
5.
Division of Applied Mathematics, Thu Dau Mot University, Binh Duong Province, Vietnam
6.
Department of Mathematics and Computer Science, University of Science, Ho Chi Minh City, Vietnam, Vietnam National University, Ho Chi Minh City, Vietnam

Received: 01 May 2020 Revised: 01 July 2020 Published: 25 August 2020
35K55, 35K70, 35K92, 47A52, 47J06

In this paper, we investigate a final boundary value problem for a class of fractional with parameter $ \beta $ pseudo-parabolic partial differential equations with nonlinear reaction term. For $ 0<\beta < 1, $ the solution is regularity-loss, we establish the well-posedness of solutions. In the case that $ \beta >1 $, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.
- Final value problem,
- pseudo-parabolic eqaution,
- regularization,
- ill-posed problem,
- estimation
Citation: Vo Van Au, Hossein Jafari, Zakia Hammouch, Nguyen Huy Tuan. On a final value problem for a nonlinear fractional pseudo-parabolic equation[J]. Electronic Research Archive, 2021, 29(1): 1709-1734. doi: 10.3934/era.2020088

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Abstract

In this paper, we investigate a final boundary value problem for a class of fractional with parameter $ \beta $ pseudo-parabolic partial differential equations with nonlinear reaction term. For $ 0<\beta < 1, $ the solution is regularity-loss, we establish the well-posedness of solutions. In the case that $ \beta >1 $, it has a feature of regularity-gain. Then, the instability of a mild solution is proved. We introduce two methods to regularize the problem. With the help of the modified Lavrentiev regularization method and Fourier truncated regularization method, we propose the regularized solutions in the cases of globally or locally Lipschitzian source term. Moreover, the error estimates is established.

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