ON THE SOLVABILITY OF DIRECT AND INVERSE PROBLEMS FOR A CLASS OF DEGENERATE PARABOLIC EQUATIONS WITH INVOLUTION

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Abstract

In this paper, for degenerate diffusion equations with involution, the solvability of the direct and inverse problems for determining the right-hand side is studied. The equation with a fractional derivative in the Caputo sense is considered. The elliptic part of the studied equation involves a nonlocal analogue of the Laplace operator with a coefficient depending on the time variable. By studying these problems with respect to the time variable, we obtain a one-dimensional degenerate equation with a fractional Caputo derivative. The solution of this equation is expressed by a special function of the Kilbas-Saigo type. Similarly, for the spatial variable, we obtain a spectral problem for the nonlocal Laplace operator with the Dirichlet boundary condition. We explicitly find the eigenfunctions and eigenvalues of this problem and show the completeness of the system of eigenfunctions in space. Using the classical Fourier method, solutions to the problems under consideration are sought in the form of expansions in a series of eigenfunctions. The absolute and uniform convergence of the series, the possibility of their differentiation term by term in all variables and the absolute and uniform convergence of the differentiated series are proved. The main statements concerning the problems considered are presented in the form of existence and uniqueness theorems.