Geodesic
On a solvability of the nonlinear inverse problem for the hyperbolic equation
Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 16 (2016) no. 1, pp. 106-129

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We investigate the nonlinear inverse problem with an unknown time depended coefficient for a second-order hyperbolic equation. The problem is stated as follows: it is required to find a solution and an unknown time depended coefficient at the solution in the equation. In the article we prove the theorems of the existence and the uniqueness of the problem’s regular solution. To study solvability of the inverse problem, we realize a conversion from the initial inverse problem to a some direct nonlocal supplementary problem for a high-order hyperbolic equation. We prove the solvability of the supplementary problem. Then we realize a conversion to the first problem again and as a result we receive the solvability of the inverse problem.
Keywords: inverse problem, hyperbolic equation, weighted equation, the method of continuation on a parameter, the fixed point theorem, cut-off functions, the method of regularization, nonlocal problem. 
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     author = {R. R. Safiullova},
     title = {On a solvability of the nonlinear inverse problem for the hyperbolic equation},
     journal = {Sibirskij \v{z}urnal \v{c}istoj i prikladnoj matematiki},
     pages = {106--129},
     publisher = {mathdoc},
     volume = {16},
     number = {1},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VNGU_2016_16_1_a6/}
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R. R. Safiullova. On a solvability of the nonlinear inverse problem for the hyperbolic equation. Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 16 (2016) no. 1, pp. 106-129. http://geodesic.mathdoc.fr/item/VNGU_2016_16_1_a6/