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

## TL;DR

This paper develops a Carleman estimate for an anisotropic hyperbolic equation with memory, applies it to derive observability inequalities, and uses these results to solve an inverse source problem with stability in viscoelasticity models.

## Contribution

It introduces a new Carleman estimate for anisotropic hyperbolic equations with memory and applies it to inverse problems and controllability in viscoelasticity.

## Key findings

- Established a Carleman estimate with boundary data.
- Derived an observability inequality for the initial state.
- Proved Lipschitz stability for the inverse source problem.

## Abstract

We consider an anisotropic hyperbolic equation with memory term: $$ \partial_t^2 u(x,t) = \sum_{i,j=1}^n \partial_i(a_{ij}(x)\partial_ju) + \int^t_0 \sum_{| \alpha| \le 2} b_{\alpha}(x,t,\eta)\partial_x^{\alpha}u(x,\eta) d\eta + F(x,t) $$ 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 $\partial_tu(\cdot,0) = 0$ 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.