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

Abstract

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.