  We consider the stochastic heat equation of the following form
\frac{\partial}{\partial t}u_t(x) = (\sL u_t)(x) +b(u_t(x)) +
\sigma(u_t(x))\dot{F}_t(x)\quad \text{for}t>0, x\in \R^d, where $\sL$ is the
generator of a L\'evy process and $\dot{F}$ is a spatially-colored, temporally
white, gaussian noise. We will be concerned mainly with the long-term behavior
of the mild solution to this stochastic PDE. For the most part, we work under
the assumptions that the initial data $u_0$ is a bounded and measurable
function and $\sigma$ is nonconstant and Lipschitz continuous. In this case, we
find conditions under which the preceding stochastic PDE admits a unique
solution which is also \emph{weakly intermittent}. In addition, we study the
same equation in the case that $\mathcal{L}u$ is replaced by its
massive/dispersive analogue $\mathcal{L}u-\lambda u$ where $\lambda\in\R$.
Furthermore, we extend our analysis to the case that the initial data $u_0$ is
a measure rather than a function. As it turns out, the stochastic PDE in
question does not have a mild solution in this case. We circumvent this problem
by introducing a new concept of a solution that we call a \emph{temperate
solution}, and proceed to investigate the existence and uniqueness of a
temperate solution. We are able to also give partial insight into the long-time
behavior of the temperate solution when it exists and is unique. Finally, we
look at the linearized version of our stochastic PDE, that is the case when
$\sigma$ is identically equal to one [any other constant works also].In this
case, we study not only the existence and uniqueness of a solution, but also
the regularity of the solution when it exists and is unique.
