Ricerc@Sapienza

On the basis of the Carleman estimate for the parabolic equation, we prove a Carleman estimate for the integro-differential operator
$\partial_t-\triangle+\int_0^t K(x,t,r)\triangle\ dr$
where the integral kernel has a behaviour like a weakly singular one.
In the proof we consider the integral term as a perturbation. The crucial point is
a special choice of the time factor of the weight function.

We consider an anisotropic hyperbolic equation with memory term: ?t2u(x,t)=?i,j=1n?i(aij(x)?ju)+?0t?|?|?2b?(x,t,?)?x?u(x,?)d?+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 $R(x, t)\,f(x)$ .