An inverse problem for the pseudo-parabolic equation with p-Laplacian

Stanislav Nikolaevich Antontsev; Serik Ersultanovich Aitzhanov; Guzel Rashitkhuzhakyzy Ashurova

doi:10.3934/eect.2021005

Article Contents

2022, Volume 11, Issue 2: 399-414. Doi: 10.3934/eect.2021005

This issue Previous Article Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains Next Article

$ (\omega,\mathbb{T}) $

-periodic solutions of impulsive evolution equations

An inverse problem for the pseudo-parabolic equation with p-Laplacian

Stanislav Nikolaevich Antontsev^1,,
Serik Ersultanovich Aitzhanov^2,3, , and
Guzel Rashitkhuzhakyzy Ashurova^2,

1.
Lavrentyev Institute of Hydrodynamics of SB RAS, Novosibirsk, Russia
2.
Al-Farabi Kazakh National University, Almaty, Kazakhstan
3.
Institute of Mathematics and Mathematical Modeling, Almaty, Kazakhstan

^* Corresponding author: aitzhanovserik81@gmail.com (Serik Ersultanovich Aitzhanov)
^* Corresponding author: aitzhanovserik81@gmail.com (Serik Ersultanovich Aitzhanov)

Received: June 30, 2020

Revised: September 30, 2020

Early access: January 2021

Published: April 2022

The first author is supported by the RSF, Russia, grant no. 19-11-00069 (50 percent of all results, Lemma 2.1, Theorem 3.4, Passage to the limit). The second and third authors were financially support by the Ministry of education and science of the Republic of Kazakhstan, grant no. AP08052425 (50 percent of all results, Introduction, Theorems 3.2, 5.1, 6.1)

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Abstract

In this article, we study the inverse problem of determining the right side of the pseudo-parabolic equation with a p-Laplacian and nonlocal integral overdetermination condition. The existence of solutions in a local and global time to the inverse problem is proved by using the Galerkin method. Sufficient conditions for blow-up (explosion) of the local solutions in a finite time are derived. The asymptotic behavior of solutions to the inverse problem is studied for large values of time. Sufficient conditions are obtained for the solution to disappear (vanish to identical zero) in a finite time. The limits conditions that which ensure the appropriate behavior of solutions are considered.

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Mathematics Subject Classification: Primary: 35R30, 35K92, 35J99, 35B44; Secondary: 35B40.

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