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 $u(x,T)=βu(x,0)+φ(x)$, 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 $β\ge 1,$ or $β<0$, then the problem is well-posed; if $β\in (0,1)$, 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.