  In this article, we identify the necessary and sufficient conditions for the
existence of a random field solution for some linear s.p.d.e.'s of parabolic
and hyperbolic type. These equations rely on a spatial operator $\cL$ given by
the $L^2$-generator of a $d$-dimensional L\'evy process $X=(X_t)_{t \geq 0}$,
and are driven by a spatially-homogeneous Gaussian noise, which is fractional
in time with Hurst index $H>1/2$. As an application, we consider the case when
$X$ is a $\beta$-stable process, with $\beta \in (0,2]$. In the parabolic case,
we develop a connection with the potential theory of the Markov process
$\bar{X}$ (defined as the symmetrization of $X$), and we show that the
existence of the solution is related to the existence of a "weighted"
intersection local time of two independent copies of $\bar{X}$.
