A non-local problem for mixed type equations involving the Caputo fractional derivative

This work is devoted to the study of a non-local problem for an equation of mixed type, which for 0<t< is a subdiffusion equation with a fractional derivative in the Caputo sense and for -<t<0 is a hyperbolic type equation with a classical derivative. The Laplace operator is involved in the elliptic part of the equations. A non-local condition has the form u (x, -) -u (x, ) = (x), x, where (x) is a given function. Using the Fourier method, conditions were found on the function (x) and on the right-hand side of the equation, under which a solution to the problem exists and it is unique.