Research output: Contribution to journal › Article › peer-review

**Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations.** / Duc, Nguyen Van; Hào, Dinh Nho; Shishlenin, Maxim.

Research output: Contribution to journal › Article › peer-review

Duc, NV, Hào, DN & Shishlenin, M 2024, 'Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations', *Journal of Inverse and Ill-Posed Problems*, vol. 32, no. 1, pp. 9-20. https://doi.org/10.1515/jiip-2023-0046

Duc, N. V., Hào, D. N., & Shishlenin, M. (2024). Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations. *Journal of Inverse and Ill-Posed Problems*, *32*(1), 9-20. https://doi.org/10.1515/jiip-2023-0046

Duc NV, Hào DN, Shishlenin M. Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations. Journal of Inverse and Ill-Posed Problems. 2024 Feb 1;32(1):9-20. doi: 10.1515/jiip-2023-0046

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title = "Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations",

abstract = "Let X be a Banach space with norm ∥ · ∥. Let A: D ⊂ (A) → X → X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ϵ > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u: [0,T] → X satisfying ut + Au = 0, 0 < t < T, ∥u (T) - φ ∥ ≤ ϵ, for φ in X, is regularized by the generalized Sobolev equation uαt + Aα uα = 0, 0 < t < T, uα (T) = φ, where 0 < α < 1 and Aα = A (I + αAb)-1 with b ≥ 1. Error estimates of the method with respect to the noise level are proved.",

keywords = "Backward parabolic equations, Sobolev equation, ill-posed problems, regularization",

author = "Duc, {Nguyen Van} and H{\`a}o, {Dinh Nho} and Maxim Shishlenin",

note = "The second author was supported by VAST project QTRU01.11/20-21. Публикация для корректировки.",

year = "2024",

month = feb,

day = "1",

doi = "10.1515/jiip-2023-0046",

language = "English",

volume = "32",

pages = "9--20",

journal = "Journal of Inverse and Ill-Posed Problems",

issn = "0928-0219",

publisher = "Walter de Gruyter GmbH",

number = "1",

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TY - JOUR

T1 - Regularization of backward parabolic equations in Banach spaces by generalized Sobolev equations

AU - Duc, Nguyen Van

AU - Hào, Dinh Nho

AU - Shishlenin, Maxim

N1 - The second author was supported by VAST project QTRU01.11/20-21. Публикация для корректировки.

PY - 2024/2/1

Y1 - 2024/2/1

N2 - Let X be a Banach space with norm ∥ · ∥. Let A: D ⊂ (A) → X → X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ϵ > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u: [0,T] → X satisfying ut + Au = 0, 0 < t < T, ∥u (T) - φ ∥ ≤ ϵ, for φ in X, is regularized by the generalized Sobolev equation uαt + Aα uα = 0, 0 < t < T, uα (T) = φ, where 0 < α < 1 and Aα = A (I + αAb)-1 with b ≥ 1. Error estimates of the method with respect to the noise level are proved.

AB - Let X be a Banach space with norm ∥ · ∥. Let A: D ⊂ (A) → X → X be an (possibly unbounded) operator that generates a uniformly bounded holomorphic semigroup. Suppose that ϵ > 0 and T > 0 are two given constants. The backward parabolic equation of finding a function u: [0,T] → X satisfying ut + Au = 0, 0 < t < T, ∥u (T) - φ ∥ ≤ ϵ, for φ in X, is regularized by the generalized Sobolev equation uαt + Aα uα = 0, 0 < t < T, uα (T) = φ, where 0 < α < 1 and Aα = A (I + αAb)-1 with b ≥ 1. Error estimates of the method with respect to the noise level are proved.

KW - Backward parabolic equations

KW - Sobolev equation

KW - ill-posed problems

KW - regularization

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U2 - 10.1515/jiip-2023-0046

DO - 10.1515/jiip-2023-0046

M3 - Article

VL - 32

SP - 9

EP - 20

JO - Journal of Inverse and Ill-Posed Problems

JF - Journal of Inverse and Ill-Posed Problems

SN - 0928-0219

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