Global Lipschitz stability in an inverse hyperbolic problem by interior observations

For the solution u ( p ) = u ( p )( x , t ) to ∂ t 2 u ( x , t )-Δ u ( x , t )- p ( x ) u ( x , t ) = 0 in Ω × (0, T ) and (∂ u /∂ν)| ∂Ω × (0, T ) = 0 with given u (·,0) and ∂ t u (·,0), we consider an inverse problem of determining p ( x ), x ∊Ω, from data u |ω × (0, T ) . Here Ω⊂ n , n = 1,2,3, is a bounded domain, ω is a sub-domain of Ω and T >0. For suitable ω⊂Ω and T >0, we prove an upper and lower estimate of Lipschitz type between || p - q || L 2 (Ω) and ||∂ t ( u ( p )- u ( q ))|| L 2 (ω × (0, T )) + ||∂ t 2 ( u ( p )- u ( q ))|| L 2 (ω × (0, T )) .