Inverse problems of parameter recovery in differential equations with multiple characteristics

Abstract

The work is devoted to the study of the solvability in Sobolev spaces of nonlinear inverse coefficient problems for differential equations of the third order with multiple characteristics.In this paper, alongside with finding the solution of one or another differential equation, it is also required to find one or more coefficients of the equation itself for us to name them inverse coefficient problems. Various aspects of the theory of inverse coefficient problems for differential equations are well covered in the world literature - see monographs ~[1-17]. At the same time, it should be noted that there are not many works devoted to the study of the solvability of inverse problems for differential equations with multiple characteristics - we can only name the works ~[17-19]. A distinctive feature of the problems studied in this paper is that the unknown coefficient is a numerical parameter, and not a function of certain independent variables. Similar problems were studied earlier, but only for classical parabolic, hyperbolic and elliptic equations - see papers ~[20-29]. The inverse problems of determination for differential equations with multiple characteristics, together with the solution of numerical parameters, which are the coefficients of the equation itself, have not been previously studied. It should be noted that differential equations with constant coefficients are often obtained by mathematical modeling of processes taking place in a homogeneous medium - see papers ~[30, 31]. If in this case the coefficients characterizing certain properties of the environment are unknown quantities, then we will automatically obtain inverse problems with unknown parameters.All constructions and arguments in this work will be carried out using Lebesgue spaces $L_p$ and Sobolev spaces $W^l_p$. The necessary information about functions from these spaces can be found in monographs ~[32-34].