  In this paper, we consider the prediction of the helium concentrations as
function of a spatially variable source term perturbed by fractional Brownian
motion. For the direct problem, we show that it is well-posed and has a unique
mild solution under some conditions. For the inverse problem, the uniqueness
and the instability are given. In the meanwhile, we determine the statistical
properties of the source from the expectation and covariance of the final-time
data u(r,T). Finally, numerical implements are given to verify the
effectiveness of the proposed reconstruction.
