# American Institute of Mathematical Sciences

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doi: 10.3934/eect.2021005

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

 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)

Received  July 2020 Revised  October 2020 Published  January 2021

Fund Project: 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)

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.

Citation: Stanislav Nikolaevich Antontsev, Serik Ersultanovich Aitzhanov, Guzel Rashitkhuzhakyzy Ashurova. An inverse problem for the pseudo-parabolic equation with p-Laplacian. Evolution Equations & Control Theory, doi: 10.3934/eect.2021005
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