  We consider the nonparametric estimation problem of time-dependent
multivariate functions observed in a presence of additive cylindrical Gaussian
white noise of a small intensity. We derive minimax lower bounds for the
$L^2$-risk in the proposed spatio-temporal model as the intensity goes to zero,
when the underlying unknown response function is assumed to belong to a ball of
appropriately constructed inhomogeneous time-dependent multivariate functions,
motivated by practical applications. Furthermore, we propose both non-adaptive
linear and adaptive non-linear wavelet estimators that are asymptotically
optimal (in the minimax sense) in a wide range of the so-constructed balls of
inhomogeneous time-dependent multivariate functions. The usefulness of the
suggested adaptive nonlinear wavelet estimator is illustrated with the help of
simulated and real-data examples.
