An inverse problem with data on the part of the boundary

Let ut = ∇2u − q(x)u ≔ Lu in D × [0, ∞), where D ⊂ R3 is a bounded domain with a smooth connected boundary S, and q(x) ∈ L2(S) is a real-valued function with compact support in D. Assume that u(x, 0) = 0, u = 0 on S1 ⊂ S, u = a(s, t) on S2 = S⧹S1, where a(s, t) = 0 for t > T, a(s, t) ≢ 0, a ∈ C1([0, T]; H3/2(S2)) is arbitrary. the extra data u N ∣ S 2 = b ( s , t ) , for each a ∈ C1([0, T]; H3/2(S2)), where N is the outer normal to S, one can find q(x) uniquely. A similar result is obtained for the heat equation u t = L u ≔ ∇ · ( a ∇ u ) . results are based on new versions of Property C.