Geodesic
An existence theorem for an inverse problem for a semilinear hyperbolic system
Differencialʹnye uravneniâ, Tome 40 (2004) no. 9, pp. 1155-1165.

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In the present paper, we consider a semilinear hyperbolic system with coefficients depending on different solution components. We study the inverse problem of finding one of the coefficients on the basis of additional information on the solution. The proof of the existence theorem for the inverse problem is based on the reduction of the latter to a nonlinear operator equation, which is a nonlinear integro-differential equation for the unknown coefficient. This approach was used in [1] to prove the existence of a solution of the inverse problem for a quasilinear hyperbolic equation. Inverse problems for quasilinear hyperbolic equations and systems were also considered in [2–6] and other papers. 
@article{DE_2004_40_9_a0,
     author = {A. M. Denisov},
     title = {An existence theorem for an inverse problem for a semilinear hyperbolic system},
     journal = {Differencialʹnye uravneni\^a},
     pages = {1155--1165},
     publisher = {mathdoc},
     volume = {40},
     number = {9},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DE_2004_40_9_a0/}
}
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A. M. Denisov. An existence theorem for an inverse problem for a semilinear hyperbolic system. Differencialʹnye uravneniâ, Tome 40 (2004) no. 9, pp. 1155-1165. http://geodesic.mathdoc.fr/item/DE_2004_40_9_a0/