Unknown coefficient problem for mixed equation of parabolic-hyperbolic type with non-local boundary conditions on characteristics

D. K. Durdiev<sup>ab

a</sup> Bukhara Branch of Romanovskii Institute of Mathematics, Uzbekistan Academy of Sciences, M. Ikbal Str. 11, 200100, Bukhara, Uzbekistan
^b Bukhara State University, M. Ikbal Str. 11, 200100, Bukhara, Uzbekistan

Abstract: For an equation of a mixed parabolic-hyperbolic type with a characteristic line of type change, we study the inverse problem associated with the search for an unknown coefficient at the lowest term of the parabolic equation. In the direct problem, we consider an analog of the Tricomi problem for this equation with a nonlocal condition on the characteristics in the hyperbolic part and the Dirichlet condition in the parabolic part of the domain. In order to determine the unknown coefficient by the solution on the parabolic part of the domain, the integral overdetermination condition is proposed. Global results on the unique solvability of the inverse problem in the sense of the classical solution are proved.

Keywords: parabolic-hyperbolic equation, characteristic, Green's function, inverse problem, contraction principle mapping.

UDC: 517.958

MSC: 35A01, 35A02, 35L02, 35L03, 35R03

Received: 12.09.2023

Language: English

 English version: Ufa Mathematical Journal, 2024, 16:2, 81–88