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The asymptotic behavior of the solution of an inverse problem for the pseudoparabolic equation

  • Anna S. Lyubanova and Aleksander V. Velisevich EMAIL logo
Published/Copyright: September 30, 2022
Journal of Inverse and Ill-posed Problems
From the journal Journal of Inverse and Ill-posed Problems Volume 31 Issue 3

Abstract

We establish the stabilization of the strong solution ( u ⁢ ( t , x ) , k ⁢ ( t ) ) to the inverse problem for a pseudoparabolic equation ( u + η ⁢ M ⁢ u ) t + M ⁢ u + k ⁢ ( t ) ⁢ u = f with an unknown coefficient k ⁢ ( t ) to the solution ( u ∞ , k ∞ ) of the appropriate stationary inverse problem. The operator M = - div ⁡ ( ℳ ⁢ ( x ) ⁢ ∇ ) + m ⁢ ( x ) ⁢ I is supposed to be elliptic and self-adjoint. The asymptotic behavior of the solution ( u ⁢ ( t , x ) , k ⁢ ( t ) ) is investigated as t → + ∞ .

Keywords: Inverse problems for PDE; boundary value problems; second-order elliptic equations; pseudoparabolic equation; asymptotic behavior of solutions
MSC 2010: 35B40; 35G16; 35J25; 35K70; 35R30

Funding source: Russian Foundation for Basic Research

Award Identifier / Grant number: 18-31-00019

Award Identifier / Grant number: 20-31-90053

Funding source: Ministry of Science and Higher Education of the Russian Federation

Award Identifier / Grant number: 075-02-2020-1534/1

Funding statement: This research is financially supported by a grant from Russian Foundation for Basic Research (Grant nos. 18-31-00019 and 20-31-90053). This work is also supported by the Krasnoyarsk Mathematical Center and financed by the Ministry of Science and Higher Education of the Russian Federation in the framework of the establishment and development of regional Centers for Mathematics Research and Education (Agreement no. 075-02-2020-1534/1).

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Received: 2021-06-26
Revised: 2021-09-22
Accepted: 2022-06-12
Published Online: 2022-09-30
Published in Print: 2023-06-01

© 2022 Walter de Gruyter GmbH, Berlin/Boston

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  2. Lipschitz stability in inverse source problems for degenerate/singular parabolic equations by the Carleman estimate
  3. The asymptotic behavior of the solution of an inverse problem for the pseudoparabolic equation
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https://doi.org/10.1515/jiip-2019-0045
Keywords for this article
Inverse problems for PDE; boundary value problems; second-order elliptic equations; pseudoparabolic equation; asymptotic behavior of solutions

Articles in the same Issue

  1. Frontmatter
  2. Lipschitz stability in inverse source problems for degenerate/singular parabolic equations by the Carleman estimate
  3. The asymptotic behavior of the solution of an inverse problem for the pseudoparabolic equation
  4. On uniqueness and nonuniqueness for internal potential reconstruction in quantum fields from one measurement II. The non-radial case
  5. Adaptive Runge–Kutta regularization for a Cauchy problem of a modified Helmholtz equation
  6. On finding a penetrable obstacle using a single electromagnetic wave in the time domain
  7. Direct and inverse problems for time-fractional heat equation generated by Dunkl operator
  8. Agent-based mathematical model of COVID-19 spread in Novosibirsk region: Identifiability, optimization and forecasting
  9. Certain inverse uniqueness from the quotients of scattering coefficients
  10. On recovery of an unbounded bi-periodic interface for the inverse fluid-solid interaction scattering problem
  11. Approximate Lipschitz stability for phaseless inverse scattering with background information
  12. The mean field games system: Carleman estimates, Lipschitz stability and uniqueness
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