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

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85167398066&origin=inward&txGid=ebdeaf6af44ed6d0962aa599d145e874

UR - https://www.mendeley.com/catalogue/b2d0d74f-56e5-3a0c-90e1-3daf1fdff201/

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

IS - 1

ER -