We consider the stochastic heat equation of the following form:
∂
∂
t
u
t
(
x
)
=
(
L
u
t
)
(
x
)
+
b
(
u
t
(
x
)
)
+
σ
(
u
t
(
x
)
)
F
˙
t
(
x
)
for
t
>
0
,
x
∈
R
d
,
\begin{equation*} \frac {\partial }{\partial t}u_t(x) = (\mathcal {L} u_t)(x) +b(u_t(x)) + \sigma (u_t(x))\dot {F}_t(x)\quad \text {for }t>0,\ x\in \mathbf {R}^d, \end{equation*}
where
L
\mathcal {L}
is the generator of a Lévy process and
F
˙
\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
u_0
is a bounded and measurable function and
σ
\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 weakly intermittent. In addition, we study the same equation in the case where
L
u
\mathcal {L}u
is replaced by its massive/dispersive analogue
L
u
−
λ
u
\mathcal {L}u-\lambda u
, where
λ
∈
R
\lambda \in \mathbf {R}
. We also accurately describe the effect of the parameter
λ
\lambda
on the intermittence of the solution in the case where
σ
(
u
)
\sigma (u)
is proportional to
u
u
[the “parabolic Anderson model”].
We also look at the linearized version of our stochastic PDE, that is, the case where
σ
\sigma
is identically equal to one [any other constant also works]. 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.