  We consider the linear stochastic heat and wave equations with generalized
Gaussian noise that is white in time and spatially correlated. Under the
assumption that the homogeneous spatial correlation $f$ satisfies some mild
conditions, we show that the solutions to the linear stochastic heat and wave
equations exhibit tall peaks in macroscopic scales, which means they are
macroscopically multi-fractal. We compute the macroscopic Hausdorff dimension
of the peaks for Gaussian random fields with vanishing correlation and then
apply this result to the solution of the linear stochastic heat and wave
equations. We also study the spatio-temporal multi-fractality of the linear
stochastic heat and wave equations. Our result is an extension of Khoshnevisan,
Kim, and Xiao \cite{KKX,KKX2} and Kim \cite{K} to a more general class of the
linear stochastic partial differential equations and Gaussian random fields.
