Identifying initial condition of the Rayleigh-Stokes problem with random noise

We investigate a backward problem for the Rayleigh-Stokes problem, which aims to determine the initial status of some physical field such as temperature for slow diffusion from its present measurement data. This problem is well-known to be ill-posed because of the rapid decay of the forward process. We construct a regularized solution using the filter regularization method in the Gaussian random noise. Under some a priori assumptions on the exact solution, we establish the expectation between the exact solution and the regularized solution in the L ² and H ^m norms.