Some classes of inverse problems for mixed type equations of second order

  • Jamalov Sirozhiddin Z., siroj63@mail.ru Academy of Sciences of the Republic of Uzbekistan, Institute of Mathematics, 81 M. Ulugbek Street, Akademgorodok, Tashkent 100170, Uzbekistan
  • Pyatkov Sergey G., pyatkov@math.nsc.ru Yugra State University, 16 Chekhov Street, Khanty-Mansyisk 628012, Russia; Sobolev Institute of Mathematics, 4 Acad. Koptyug Avenue, Novosibirsk 630090, Russia
Keywords: mixed type equation, inverse problem, source function, gas dynamics

Abstract

In the Sobolev spaces, we consider the well-posedness questions for the inverse problem of recovering the source function of a mixed type equation of second order. The overdetermination conditions are the values of a solution on a collection of planes of dimension n − 1. The unknowns occurring in the right-hand side depend on time and n − 1 unknown space variables. Under certain natural conditions on the data of the problem, we obtain existence and uniqueness theorems for generalized solutions to this problem. The conditions on the data almost coincide with those ensuring solvability of the direct problem. The parameter continuation method and a priori estimates are used to validate the results. The method allows us to generalize the results to the case of smoother data and regular solutions.

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How to Cite
Jamalov, S. and Pyatkov, S. ( ) “Some classes of inverse problems for mixed type equations of second order”, Mathematical notes of NEFU, 25(4), pp. 3-14. doi: https://doi.org/10.25587/SVFU.2018.100.20550.
Section
Mathematics