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**Inverse boundary value problem for the linearized Benney-Luke equation with nonlocal conditions.**
*(Russian.
English summary)*
Zbl 1442.35548

Summary: The paper investigates the solvability of an inverse boundary-value problem with an unknown coefficient and the right-hand side, depending on the time variable, for the linearized Benney-Luke equation with non-self-adjoint boundary and additional integral conditions. The problem is considered in a rectangular domain. A definition of the classical solution of the problem is given. First, we consider an auxiliary inverse boundary-value problem and prove its equivalence (in a certain sense) to the original problem. To investigate the auxiliary inverse boundary-value problem, the method of separation of variables is used. By applying the formal scheme of the variable separation method, the solution of the direct boundary problem (for a given unknown function) is reduced to solving the problem with unknown coefficients. Then, the solution of the problem is reduced to solving a certain countable system of integro-differential equations for the unknown coefficients. In turn, the latter system of relatively unknown coefficients is written as a single integro-differential equation for the desired solution. Next, using the corresponding additional conditions of the inverse auxiliary boundary-value problem, to determine the unknown functions, we obtain a system of two nonlinear integral equations. Thus, the solution of an auxiliary inverse boundary-value problem is reduced to a system of three nonlinear integro-differential equations with respect to unknown functions. A special type of Banach space is constructed. Further, in a ball from a constructed Banach space, with the help of contracted mappings, we prove the solvability of a system of nonlinear integro-differential equations, which is also the unique solution to the auxiliary inverse boundary-value problem. Finally, using the equivalence of these problems the existence and uniqueness of the classical solution of the original problem are proved.

### MSC:

35R30 | Inverse problems for PDEs |

35G16 | Initial-boundary value problems for linear higher-order PDEs |

### Keywords:

inverse boundary value problem; Benney-Luke equation; existence; uniqueness of classical solution
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\textit{Y. T. Megraliev} and \textit{B. K. Velieva}, Vestn. Udmurt. Univ., Mat. Mekh. Komp'yut. Nauki 29, No. 2, 166--182 (2019; Zbl 1442.35548)

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