Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case. (22nd November 2017)

• Record Type: Journal Article
• Authors: Loreti, Paola; Sforza, Daniela; Yamamoto, Masahiro
• Is Part Of: Inverse problems. Volume 33:Number 12(2017:Dec.)
• Publication Date: 2017-11-22
• Subjects: inverse source problem -- viscoelasticity -- anisotropic media -- Carleman estimate; Inverse problems (Differential equations) -- Periodicals; 515.357
• DOI: 10.1088/1361-6420/aa96c1
• Languages: English
• ISSNs: 0266-5611

Abstract

Abstract: We consider an anisotropic hyperbolic equation with memory term: ∂ t 2 u ( x, t ) = ∑ i, j = 1 n ∂ i ( a i j ( x ) ∂ j u ) + ∫ 0 t ∑ | α | ⩽ 2 b α ( x, t, η ) ∂ x α u ( x, η ) d η + R ( x, t ) f ( x ) for x ∈ Ω and t ∈ ( 0, T ), which is a simplified model equation for viscoelasticity. The main result is a both-sided Lipschitz stability estimate for an inverse source problem of determining a spatial varying factor f ( x ) of the force term R ( x, t ) f ( x ) . The proof is based on a Carleman estimate and due to the anisotropy, the existing transformation technique does not work and we introduce a new transformation of u in order to treat the integral terms.