  Let $\boldsymbol{X}(t)=(X_1(t),\ldots,X_d(t))$ be a Gaussian vector process
and $g(t)$ be a continuous function. The asymptotics of distribution of
$\left\|\boldsymbol{X}(t)\right\|_p$, the $L^p$ norm for Gaussian
finite-dimensional vector, have been investigated in numerous literatures. In
this contribution we are concerned with the exact tail asymptotics of
$\left\|\boldsymbol{X}(t)\right\|^c_p,\ c>0, $ with trend $g(t)$ over $[0,T]$.
Both scenarios that $\boldsymbol{X}(t)$ is locally stationary and
non-stationary are considered. Important examples include $\sum_{i=1}^d
\left|X_i(t)\right|+g(t)$ and chi-square processes with trend, i.e.,
$\sum_{i=1}^d X_i^2(t)+g(t)$. These results are of interest in applications in
engineering, insurance and statistics, etc.
