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 $0<t<\beta$ is a subdiffusion equation with a fractional derivative in the Caputo sense and for $-\alpha<t<0$ 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 $u(x,-\alpha)-u(x,\beta)=\varphi(x),$ $x\in\Omega,$ where $\varphi(x) $ is a given function. Using the Fourier method, conditions were found on the function $\varphi(x) $ and on the right-hand side of the equation, under which a solution to the problem exists and it is unique.