On some classes of inverse problems for parabolic and elliptic equations

Pyatkov, S. G.; Tsybikov, B. N.

doi:10.1007/s00028-010-0087-6

Cited by 13 publications

(6 citation statements)

References 25 publications

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“…2 in [25]; it is similar to that in the proof of Theorem 1.1 in [9] or in the proof of Theorem 3.1 in [26].…”

Section: Preliminariesmentioning

confidence: 64%

See 1 more Smart Citation

Parameter Identification and Control in Heat Transfer Processes

Pyatkov¹,

Goncharenko²

2017

Bulletin of the SUSU. MMP

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The article is devoted to the study of some mathematical models describing heat transfer processes. We examine an inverse problem of recovering a control parameter providing a prescribed temperature distribution at a given point of the spatial domain. The parameter is a lower order coecient depending on time in a parabolic equation. This nonlinear problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the xed point theorem. Existence and uniqueness theorems of solutions to this problem are stated and proved. Stability estimates are exposed. The main result is the global (in time) existence of solutions under some natural conditions of the data. The proofs rely on the maximum principle. The main functional spaces used are the Sobolev spaces.

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“…2 in [25]; it is similar to that in the proof of Theorem 1.1 in [9] or in the proof of Theorem 3.1 in [26].…”

Section: Preliminariesmentioning

confidence: 64%

“…Generally speaking our reference to Theorem 3.1 in [26] is not exact, since the case of dierent boundary conditions on dierent connectedness components of the boundary is not treated there. However the proof of Theorem 3.1 remains valid in this case as well, since it is based on a partition of unity and local considerations.…”

Section: Preliminariesmentioning

confidence: 99%

Parameter Identification and Control in Heat Transfer Processes

Pyatkov¹,

Goncharenko²

2017

Bulletin of the SUSU. MMP

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“…In view of the method, all coefficients also are independent of some spacial variables. More complete results for the problems (1)-( 3) can be found in [14][15][16][17], where the well-posedness of the inverse problems in question is established for the case of the additional data are the values of a solution on some spacial manifolds or at some collection of points. However, in these articles   ,1 c t x  except for the article [15], where     , c t x c t  in the case of the pointwise ovedetermination.…”

Section: Introductionmentioning

confidence: 99%

On Some Classes of Inverse Parabolic Problems of Recovering the Thermophysical Parameters

Pyatkov,

Soldatov

2023

MMPh

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In the article we examine the question of regular solvability in Sobolev spaces of parabolic inverse coefficient problems. A solution is sought in the class of regular solutions that has all derivatives occurring in the equation summable to some power. The overdetermination conditions are the values of a solution at some collection of points lying inside the domain. The proof is based on a priori estimates and the fixed point theorem.

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“…В одномерном случае, когда , такие линейная и нелинейная задачи были изучены в пространствах Гельдера в [1]. Можно отметить работы [2], [3], где были рассмотрены задачи вида (1), (2) в общей постановке. В данной работе при выполнении условия параболичности приводятся оценки устойчивости решений задачи (1)-(3) в пространствах Соболева и получена также локальная по времени корректность, то есть доказано существование, единственность и непрерывная зависимость решений от данных задачи.…”

Section: Introductionunclassified