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Property C and an inverse problem for a hyperbolic equation. (English) Zbl 0729.35147
Summary: Let (*) u tt -Δu+q(x,t)u=0 in D×[0,T], where DR 3 is a bounded domain with a smooth boundary D, T>d, d:=diamD, q(x,t)C([0,T], L (D)). Suppose that for every (**) u| D =f(x,t)C 1 (D×[0,T]), the value u N | D :=h(s,t) is known, where N is the outer normal to D, u solves (*) and (**) and satisfies the initial conditions u=u t =0 at t=0. Then q(x,t) is uniquely determined by the data { f,h}, fC 1 (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 D ¯×[0,T], provided that q(x,t) is known outside S.
35R30Inverse problems for PDE
35L20Second order hyperbolic equations, boundary value problems