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Property C and an inverse problem for a hyperbolic equation. (English) Zbl 0729.35147
Summary: Let (*) $u\sb{tt}-\Delta u+q(x,t)u=0$ in $D\times [0,T]$, where $D\subset R\sp 3$ is a bounded domain with a smooth boundary $\partial D$, $T>d$, $d:=diam D$, q(x,t)$\in C([0,T]$, $L\sp{\infty}(D))$. Suppose that for every (**) $u\vert\sb{\partial D}=f(x,t)\in C\sp 1(\partial D\times [0,T])$, the value $u\sb N\vert\sb{\partial D}:=h(s,t)$ is known, where N is the outer normal to $\partial D$, u solves (*) and (**) and satisfies the initial conditions $u=u\sb t=0$ at $t=0$. Then q(x,t) is uniquely determined by the data $\{$ f,h$\}$, $\forall f\in C\sp 1(\partial D\times [0,T])$ in the subset S of $D\times [0,T]$ consisting of the lines which make 45$\circ$ with the t-axis and which meet the planes $t=0$ and $t=T$ outside $\bar D\times [0,T]$, provided that q(x,t) is known outside S.

35R30Inverse problems for PDE
35L20Second order hyperbolic equations, boundary value problems
Full Text: DOI
[1] Ramm, A. G.: On completeness of the products of harmonic functions. Proc. amer. Math. soc. 99, 253-256 (1986) · Zbl 0602.35015
[2] Ramm, A. G.: Completeness of the products of solutions to PDE and uniqueness theorems in inverse scattering. Inverse problems 3, L77-L82 (1987) · Zbl 0636.35066
[3] Ramm, A. G.: J. math. Anal. appl.. 139, 302 (1989)
[4] Ramm, A. G.: A uniqueness theorem for a boundary inverse problem. Inverse problems 4, L1-L5 (1988) · Zbl 0696.35191
[5] Ramm, A. G.: A uniqueness theorem for two-parameter inversion. Inverse problems 4, L7-L10 (1988) · Zbl 0699.35250
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