Global Lipschitz stability in an inverse hyperbolic problem by interior observations

For the solution u (p) = u (p) (x, t) to partial derivative (2)(t)u, (x, t) -Deltau (x, t) - p(x)u(x, t) = 0 in Omega x (0, T) and partial derivativeu /partial derivativev\ partial derivative Omega x (0, T) = 0 with given u(., 0) and partial derivative (t)u (., 0), we consider an inverse problem of d etermining p(x), x epsilon Omega, from data u(\omegax(0,T)). Here Omega sub set of R-n, n = 1, 2, 3, is a bounded domain, omega is a sub-domain of n an d T > 0. For suitable co C n and T > 0, we prove an upper and lower estimat e of Lipschitz type between \\P - q\\(L2(Omega)) and \\partial derivative ( t)(u(p) - u(q))\\(L2(omega (0,T))) + \\partial derivative (2)(t)(u(p) - u(q ))\\(L2(omegax(0,T))).