Property C and an inverse problem for a hyperbolic equation

AbstractLet (∗) utt − Δu + q(x, t) u = 0 in D × [0, T], where D ⊂R3 is a bounded domain with a smooth boundary ∂D, T > d, d: = diam D, q(x, t) ϵ C([0, T], L∞(D)). Suppose that for every (∗∗) u¦∂D = f(x, t) ϵ C1(∂D × [0, T]), the value uN¦∂D: = h(s, t) is known, where N is the outer normal to ∂D, u solves (∗) and (∗∗) and satisfies the initial conditions u = ut = 0 at t = 0. Then q(x, t) is uniquely determined by the data {f, h}, ∀f ϵ C1(∂D × [0, T]) in the subset S of D × [0, T] consisting of the lines which make 45 ° with the t-axis and which meet the planes t = 0 and t = T outside D̄ × [0, T], provided that q(x, t) is known outside S. Here D is the closure of D