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April  2022, 11(2): 399-414. 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  April 2022 Early access  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 and Control Theory, 2022, 11 (2) : 399-414. doi: 10.3934/eect.2021005
##### References:
 [1] A. Asanov and E. R. Atamanov, An inverse problem for a pseudoparabolic integrodifferential operator equation, Siberian Mathematical Journal, 36 (1995), 645-655.  doi: 10.1007/BF02107322. [2] A. Asanov and E. R. Atamanov, Nonclassical and Inverse Problems for Pseudoparabolic Equations, De Gruyter, Berlin, 1997. doi: 10.1515/9783110900149. [3] A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter, Berlin, 2011. doi: 10.1515/9783110255294. [4] S. N. Antontsev and Kh. Khompysh, Kelvin-Voigt equation with p-Laplacian and damping term: Existence, uniqueness and blow–up, Mathematical Analysis and Applications, 446 (2017), 1255-1273.  doi: 10.1016/j.jmaa.2016.09.023. [5] S. N. Antontsev and Kh. Khompysh, Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms, Mathematical Analysis and Applications, 456 (2017), 99-116.  doi: 10.1016/j.jmaa.2017.06.056. [6] S. N. Antontsev, H. B. de Oliveira and Kh. Khompysh, Kelvin-Voigt equations perturbed by anisotropic relaxation, diffusion and damping, Mathematical Analysis and Applications, 473 (2019), 1122-1154.  doi: 10.1016/j.jmaa.2019.01.011. [7] S. N. Antontsev, H. B. de Oliveira and Kh. Khompysh, Generalized Kelvin-Voigt equations for nonhomogeneous and incompressible fluids, Communications in Mathematical Sciences, 17 (2019), 1915-1948.  doi: 10.4310/CMS.2019.v17.n7.a7. [8] S. N. Antontsev, H. B. de Oliveira and Kh. Khompysh, Kelvin-Voigt equations with anisotropic diffusion, relaxation, and damping: Blow-up and large time behavior, Asymptotic Analysis, 121 (2021), 125–157. doi: 10.3233/ASY-201597. [9] S. N. Antontsev and S. E. Aitzhanov, Inverse problem for an equation with a nonstandard growth condition, Journal of Applied Mechanics and Technical Physics, 60 (2019), 265-277.  doi: 10.1134/S0021894419020081. [10] S. E. Aitzhanov and D. T. Zhanuzakova, Behavior of solutions to an inverse problem for a quasilinear parabolic equation, Siberian Electronic Mathematical Reports, 16 (2019), 1393-1409.  doi: 10.33048/semi.2019.16.097. [11] S. N. Antontsev, J. I. Díaz and S. Shmarev, Energy Methods for Free Boundary Problems: Progress in Nonlinear Differential Equations and Their Applications, Birkhäuse Boston, Inc., Boston, 2002. doi: 10.1007/978-1-4612-0091-8. [12] B. P. Demidovič, Lectures on the Mathematical Stability Theory, Nauka, Moscow, in Russian, 1967. [13] A. Favini and A. Lorenzi, Differential Equations, Inverse and Direct Problems, Tylor and Francis Group, LLC, 2006. [14] S. I. Kabanikhin, Inverse and Ill-posed Problems: Theory and Applications, De Gruyter, Berlin, 2012. [15] A. I. Kozhanov, Inverse problems for determining boundary regimes for some equations of sobolev type, Bulletin of The South Ural State University Series-Mathematical Modelling Programming and Computer Software, in Russian, 9 (2016), 37–45. doi: 10.14529/mmp160204. [16] A. I. Kozhanov, Composite Type Equations and Inverse Problems, VSP, Netherlands, 1999. doi: 10.1515/9783110943276. [17] V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New-York-Berlin-Heidelberg, 1998. doi: 10.1007/978-1-4899-0030-2. [18] A. Sh. Lyubanova and A. Tani, An inverse problem for pseudoparabolic equation of filtration, The existence, uniqueness and regularity, Applicable Analysis, 90 (2011), 1557-1571.  doi: 10.1080/00036811.2010.530258. [19] A. Sh. Lyubanova and A. Tani, On inverse problems for pseudoparabolic and parabolic equations of filtration, Inverse Problems in Science and Engineering, 19 (2011), 1023-1042.  doi: 10.1080/17415977.2011.569712. [20] A. Sh. Lyubanova and A. Tani, An inverse problem for pseudoparabolic equation of filtration: The stabilization, Applicable Analysis, 92 (2013), 573-585.  doi: 10.1080/00036811.2011.630667. [21] A. Sh. Lyubanova and A. V. Velisevich, Inverse problems for the stationary and pseudoparabolic equations of diffusion, Applicable Analysis, 98 (2019), 1997-2010.  doi: 10.1080/00036811.2018.1442001. [22] J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod, Paris, 1969. [23] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Method for Solving Inverse Problems in Mathematical Physics, Vol. 231, Marcel Dekker: Monograths and Textbooks in Pure and Applied Mathematics, 2000. [24] S. G. Pyatkov and S. N. Shergin, On some mathematical models of filtration theory, Bulletin of The South Ural State University Series-Mathematical Modelling Programming and Computer Software, in Russian, 8 (2015), 105–116. [25] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987. [26] M. Yaman, Blow-up solution and stability to an inverse problem for a pseudo-parabolic equation, Journal of Inequalities and Applications, 2012 (2012), 1-8.  doi: 10.1186/1029-242X-2012-274.

show all references

##### References:
 [1] A. Asanov and E. R. Atamanov, An inverse problem for a pseudoparabolic integrodifferential operator equation, Siberian Mathematical Journal, 36 (1995), 645-655.  doi: 10.1007/BF02107322. [2] A. Asanov and E. R. Atamanov, Nonclassical and Inverse Problems for Pseudoparabolic Equations, De Gruyter, Berlin, 1997. doi: 10.1515/9783110900149. [3] A. B. Al'shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter, Berlin, 2011. doi: 10.1515/9783110255294. [4] S. N. Antontsev and Kh. Khompysh, Kelvin-Voigt equation with p-Laplacian and damping term: Existence, uniqueness and blow–up, Mathematical Analysis and Applications, 446 (2017), 1255-1273.  doi: 10.1016/j.jmaa.2016.09.023. [5] S. N. Antontsev and Kh. Khompysh, Generalized Kelvin–Voigt equations with p-Laplacian and source/absorption terms, Mathematical Analysis and Applications, 456 (2017), 99-116.  doi: 10.1016/j.jmaa.2017.06.056. [6] S. N. Antontsev, H. B. de Oliveira and Kh. Khompysh, Kelvin-Voigt equations perturbed by anisotropic relaxation, diffusion and damping, Mathematical Analysis and Applications, 473 (2019), 1122-1154.  doi: 10.1016/j.jmaa.2019.01.011. [7] S. N. Antontsev, H. B. de Oliveira and Kh. Khompysh, Generalized Kelvin-Voigt equations for nonhomogeneous and incompressible fluids, Communications in Mathematical Sciences, 17 (2019), 1915-1948.  doi: 10.4310/CMS.2019.v17.n7.a7. [8] S. N. Antontsev, H. B. de Oliveira and Kh. Khompysh, Kelvin-Voigt equations with anisotropic diffusion, relaxation, and damping: Blow-up and large time behavior, Asymptotic Analysis, 121 (2021), 125–157. doi: 10.3233/ASY-201597. [9] S. N. Antontsev and S. E. Aitzhanov, Inverse problem for an equation with a nonstandard growth condition, Journal of Applied Mechanics and Technical Physics, 60 (2019), 265-277.  doi: 10.1134/S0021894419020081. [10] S. E. Aitzhanov and D. T. Zhanuzakova, Behavior of solutions to an inverse problem for a quasilinear parabolic equation, Siberian Electronic Mathematical Reports, 16 (2019), 1393-1409.  doi: 10.33048/semi.2019.16.097. [11] S. N. Antontsev, J. I. Díaz and S. Shmarev, Energy Methods for Free Boundary Problems: Progress in Nonlinear Differential Equations and Their Applications, Birkhäuse Boston, Inc., Boston, 2002. doi: 10.1007/978-1-4612-0091-8. [12] B. P. Demidovič, Lectures on the Mathematical Stability Theory, Nauka, Moscow, in Russian, 1967. [13] A. Favini and A. Lorenzi, Differential Equations, Inverse and Direct Problems, Tylor and Francis Group, LLC, 2006. [14] S. I. Kabanikhin, Inverse and Ill-posed Problems: Theory and Applications, De Gruyter, Berlin, 2012. [15] A. I. Kozhanov, Inverse problems for determining boundary regimes for some equations of sobolev type, Bulletin of The South Ural State University Series-Mathematical Modelling Programming and Computer Software, in Russian, 9 (2016), 37–45. doi: 10.14529/mmp160204. [16] A. I. Kozhanov, Composite Type Equations and Inverse Problems, VSP, Netherlands, 1999. doi: 10.1515/9783110943276. [17] V. Isakov, Inverse Problems for Partial Differential Equations, Springer, New-York-Berlin-Heidelberg, 1998. doi: 10.1007/978-1-4899-0030-2. [18] A. Sh. Lyubanova and A. Tani, An inverse problem for pseudoparabolic equation of filtration, The existence, uniqueness and regularity, Applicable Analysis, 90 (2011), 1557-1571.  doi: 10.1080/00036811.2010.530258. [19] A. Sh. Lyubanova and A. Tani, On inverse problems for pseudoparabolic and parabolic equations of filtration, Inverse Problems in Science and Engineering, 19 (2011), 1023-1042.  doi: 10.1080/17415977.2011.569712. [20] A. Sh. Lyubanova and A. Tani, An inverse problem for pseudoparabolic equation of filtration: The stabilization, Applicable Analysis, 92 (2013), 573-585.  doi: 10.1080/00036811.2011.630667. [21] A. Sh. Lyubanova and A. V. Velisevich, Inverse problems for the stationary and pseudoparabolic equations of diffusion, Applicable Analysis, 98 (2019), 1997-2010.  doi: 10.1080/00036811.2018.1442001. [22] J.-L. Lions, Quelques Méthodes de Résolution des Problémes aux Limites non Linéaires, Dunod, Paris, 1969. [23] A. I. Prilepko, D. G. Orlovsky and I. A. Vasin, Method for Solving Inverse Problems in Mathematical Physics, Vol. 231, Marcel Dekker: Monograths and Textbooks in Pure and Applied Mathematics, 2000. [24] S. G. Pyatkov and S. N. Shergin, On some mathematical models of filtration theory, Bulletin of The South Ural State University Series-Mathematical Modelling Programming and Computer Software, in Russian, 8 (2015), 105–116. [25] V. G. Romanov, Inverse Problems of Mathematical Physics, VNU Science Press, Utrecht, 1987. [26] M. Yaman, Blow-up solution and stability to an inverse problem for a pseudo-parabolic equation, Journal of Inequalities and Applications, 2012 (2012), 1-8.  doi: 10.1186/1029-242X-2012-274.
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