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 ) = βu(x, 0) + ϕ(x), where β is an arbitrary real number, is proposed instead of the initial condition. If β = 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 β = 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 β, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if β ≥ 1, or β < 0, then the problem is wellposed; if β ∈ (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.

In recent years, much attention has been paid to the study of forward and inverse problems for the Rayleigh-Stokes equation in connection with the importance of this equation for applications. This equation plays an important role, in particular, in the study of the behavior of certain non-Newtonian fluids. The equation includes a fractional derivative of order α, which is used to describe the viscoelastic behavior of the flow. In this paper, we study the behavior of the solution of such equations depending on the parameter α. In particular, it is proved that for sufficiently large t the norm ||u(x, t)|| L 2 (Ω) of the solution decreases with respect to α. Moreover the inverse problem of determining the order of the derivative α is solved uniquely.Key words and phrases. The Rayleigh-Stokes problem, dependence of solution on the order of derivetive, inverse problem, determination of order of derivative, Fourier method.

A nonlocal boundary value problem for the fractional version of the Rayleigh–Stokes equation, well-known in fluid dynamics, is studied. Namely, the condition u(x,T)=βu(x,0)+φ(x), where β is an arbitrary real number, is proposed instead of the initial condition. If β=0, then we have 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 β=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 β, which separates two types of behavior of the semi-backward problem under consideration. We prove the following statements: if β≥1, or β<0, then the problem is well-posed; if β∈(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.

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