Recovery of a source term or a speed with one measurement and applications

We study the problem of recovery the source $a(t,x)F(x)$ in the wave equation in anisotropic medium with $a$ known so that $a(0,x)\not=0$ with a single measurement. We use Carleman estimates combined with geometric arguments and give sharp conditions for uniqueness. We also study the non-linear problem of recovery the sound speed in the equation $u_{tt} -c^2(x)\Delta u =0$ with one measurement. We give sharp conditions for stability, as well. An application to thermoacoustic tomography is also presented.