Geodesic
Inverse problems for hyperbolic equations: the case of unknown time-dependent coefficients
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 1, pp. 43-59
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Solvability is studied of the inverse problems of finding a solution $u(x,t)$ and the coefficients $q(t)$ of the equations \begin{align*} u_{tt}-u_{xx}+q(t)a(x,t)u_t=(x,t),\\ u_{tt}-u_{xx}+q(t)a(x,t)u=(x,t) \end{align*} In this case the overdetermination condition has the integral form $$ \int_0^1K(x,t)u(x,t)dx=\mu(t). $$ Unique existence of regular solutions is established. 
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I. R. Valitov; A. I. Kozhanov. Inverse problems for hyperbolic equations: the case of unknown time-dependent coefficients. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 6 (2006) no. 1, pp. 43-59. http://geodesic.mathdoc.fr/item/VNGU_2006_6_1_a2/

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