A Non-Local Problem for the Fractional-Order Rayleigh–Stokes Equation

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