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: $${\partial }_t^{2}u(x,t)=\sum _{i,j=1}^n{\partial }_i(a_{ij}(x){\partial }_{j}u)+{\int }_0^t\sum _{|\alpha |\le 2}{b}_{\alpha }(x,t,\eta ){\partial }_x^{\alpha }u(x,\eta )d\eta +F(x,t)$$ for $x\in \Omega$ and $t\in (0,T)$ or $\in (-T,T)$, which is a model equation for viscoelasticity. First we establish a Carleman estimate for this equation with overdetermining boundary data on a suitable lateral subboundary $\Gamma \times (-T,T)$. Second we apply the Carleman estimate to establish a both-sided estimate of $|u(\cdot ,0){|}_{H^3(\Omega )}$ by ${\partial }_{\nu }u{|}_{\Gamma \times (0,T)}$ under the assumption that ${\partial }_{t}u(\cdot ,0)=0$ and $T>0$ is sufficiently large, $\Gamma \subset \partial \Omega$ satisfies some geometric condition. Such an estimate is a kind of observability inequality and related to the exact controllability. Finally we apply the Carleman estimate to an inverse source problem of determining a spatial varying factor in $F(x,t)$ and we establish a both-sided Lipschitz stability estimate.

References in corpus (2)

• Inverse problems for two by two reaction-diffusion system using a Carleman estimate with one observation
• Carleman estimate for linear viscoelasticity equations and an inverse source problem

Cited by in corpus (3)

• Carleman estimate and an inverse source problem for the Kelvin-Voigt model for viscoelasticity
• Carleman estimate for linear viscoelasticity equations and an inverse source problem
• Source Reconstruction and Stability via Boundary Control of Abstract Viscoelastic Systems