A non-local problem for the fractional order Rayleigh-Stokes equation

A nonlocal boundary value problem for the fractional version of the well known in fluid dynamics Rayleigh-Stokes equation is studied. Namely, the condition $u(x,T)=\beta u(x,0)+\varphi(x)$, where $\beta$ is an arbitrary real number, is proposed instead of the initial condition. If $\beta=0$, then we get the inverse problem in time, called the backward problem. It is well known that the backward problem is ill-posed in the sense of Hadamard. If $\beta=1$, then the corresponding non-local problem becomes well-posed in the sense of Hadamard, and moreover, in this case a coercive estimate for the solution can be established. The aim of this work is to find values of the parameter $\beta$, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if $\beta \ge 1,$ or $\beta<0$, then the problem is well-posed; if $\beta\in (0,1)$, then depending on the eigenvalues of the elliptic part of the equation, for the existence of a solution an additional condition on orthogonality of the right-hand side of the equation and the boundary function to some eigenfunctions of the corresponding elliptic operator may emerge.