TY - JOUR

T1 - Uniqueness and stability analysis of final data inverse source problems for evolution equations

AU - Romanov, Vladimir

AU - Hasanov, Alemdar

N1 - Publisher Copyright: © 2022 Walter de Gruyter GmbH, Berlin/Boston.

PY - 2022/6/1

Y1 - 2022/6/1

N2 - 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).

AB - 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).

KW - damped wave

KW - Euler-Bernoulli and Kirchhoff equations

KW - final time output

KW - Heat

KW - inverse source problem

KW - stability

KW - stability radius

KW - uniqueness

UR - http://www.scopus.com/inward/record.url?scp=85129711650&partnerID=8YFLogxK

U2 - 10.1515/jiip-2021-0072

DO - 10.1515/jiip-2021-0072

M3 - Article

AN - SCOPUS:85129711650

VL - 30

SP - 425

EP - 446

JO - Journal of Inverse and Ill-Posed Problems

JF - Journal of Inverse and Ill-Posed Problems

SN - 0928-0219

IS - 3

ER -