We consider an anisotropic hyperbolic equation with memory term: ∂t2u(x,t)=∑i,j=1n∂i(aij(x)∂ju)+∫0t∑|α|≤2bα(x,t,η)∂xαu(x,η)dη+R(x,t)f(x) for $x \in \Omega$ and $t\in (0, T)$ , 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.

Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case / Loreti, Paola; Sforza, Daniela; Yamamoto, Masahiro. - In: INVERSE PROBLEMS. - ISSN 0266-5611. - ELETTRONICO. - 33:12(2017), p. 125014. [10.1088/1361-6420/aa96c1]

### Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case

#### Abstract

We consider an anisotropic hyperbolic equation with memory term: ∂t2u(x,t)=∑i,j=1n∂i(aij(x)∂ju)+∫0t∑|α|≤2bα(x,t,η)∂xαu(x,η)dη+R(x,t)f(x) for $x \in \Omega$ and $t\in (0, T)$ , 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.
##### Scheda breve Scheda completa
anisotropic media; Carleman estimate; inverse source problem; viscoelasticity; Theoretical Computer Science; Signal Processing; Mathematical Physics; Computer Science Applications1707 Computer Vision and Pattern Recognition; Applied Mathematics
01 Pubblicazione su rivista::01a Articolo in rivista
Carleman estimate and application to an inverse source problem for a viscoelasticity model in anisotropic case / Loreti, Paola; Sforza, Daniela; Yamamoto, Masahiro. - In: INVERSE PROBLEMS. - ISSN 0266-5611. - ELETTRONICO. - 33:12(2017), p. 125014. [10.1088/1361-6420/aa96c1]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1114722