Uniqueness and stability analysis of final data inverse source problems for evolution equations

Vladimir Romanov, Alemdar Hasanov

Результат исследования: Научные публикации в периодических изданияхстатьярецензирование

Аннотация

This article proposes a unified approach to the issues of uniqueness and Lipschitz stability for the final data inverse source problems of determining the unknown spatial load F (x) in the evolution equations. The approach is based on integral identities outlined here for the one-dimensional and multidimensional heat equations ρ (x) u t - (k (x) u x) x = F (x) G (t), (x, t) ϵ(0, ℓ) × (0, T ], and ρ (x) u t - div (k (x) ∇ u) = F (x) G (x, t), (x, t) ϵω × (0, T], ω ϵℝn, for the damped wave, and the Euler-Bernoulli beam and Kirchhoff plate equations ρ (x) u tt + μ (x) ut - (r (x) u x) x = F (x) G (x, t), ρ (x) u tt + μ (x) ut + (r (x) u xx) xx = F (x) G (t), for (x, t) ϵ(0, ℓ) × (0, T ] {(x,t)\in(0,\ell)\times(0,T]}, and ρ (x) h (x) u tt + μ (x) ut + (D (x) (u x1, x1 + ν u x2, x2)) x1, x1 + (D (x) (u x2, x2 + ν u x1, x1)) x2, x2 + 2 (1 - ν) (D (x) u x1, x2) x1, x2 = F (x) G (t), (x, t) ϵω T:= ω × (0, T), ω:= (0, ℓ 1) × (0, ℓ 2), and allows us to prove the uniqueness and stability of the solutions for all considered inverse problems under the same conditions imposed on the load G (t) or G (x, t).

Язык оригиналаанглийский
Страницы (с-по)425-446
Число страниц22
ЖурналJournal of Inverse and Ill-Posed Problems
Том30
Номер выпуска3
DOI
СостояниеОпубликовано - 1 июн 2022

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