Unique continuation for weak solutions of the wave equation plus a potential (Q1900595)

For the wave equation \[ {\partial^2 u\over \partial t^2}- \sum^{n- 1}_1 {\partial^2 u\over \partial x^2_j}= v(x, t) u(x, t) \] it is considered the problem of the uniqueness across hypersurfaces of the weak solutions. Namely, there are studied \(L^2\)- solutions with the potential \(v(x, t)\) in \(L^n_{\text{loc}}\). The studies are motivated by the applications to the boundary control and to the decay of solutions of semilinear wave equations with locally distributed damping. The main result states that if a weak solution is zero outside the cylinder \(D_R= (- T,T)\times \{x\in \mathbb{R}^{n- 1}\bigl||x|< R\}\) then it is zero everywhere.