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

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.