Excursions of a Gaussian process with variable variance above a barrier increasing to infinity

For a family of real-valued Gaussian processes $\xi_u(t)$, $t\in[0,T]$, we obtain an exact asymptotics of the probability of crossing a level $u$ as $u\to\infty$ under certain conditions on the variance and correlation. This result is applied to the investigation of excursions of a stationary zero-mean process above a barrier increasing to infinity.