Title: Dense blowup for parabolic SPDEs,
Abstract: The main result of this paper is that there are examples of stochastic
partial differential equations [hereforth, SPDEs] of the type $$ \partial_t
u=\frac12\Delta u +\sigma(u)\eta \qquad\text{on
$(0\,,\infty)\times\mathbb{R}^3$}$$ such that the solution exists and is unique
as a random field in the sense of Dalang and Walsh, yet the solution has
unbounded oscillations in every open neighborhood of every space-time point. We
are not aware of the existence of such a construction in spatial dimensions
below $3$. En route, it will be proved that there exist a large family of
parabolic SPDEs whose moment Lyapunov exponents grow at least sub exponentially
in its order parameter in the sense that there exist $A_1,\beta\in(0\,,1)$ such
that \[
\underline{\gamma}(k) :=
\liminf_{t\to\infty}t^{-1}\inf_{x\in\mathbb{R}^3}
\log\mathbb{E}\left(|u(t\,,x)|^k\right) \ge A_1\exp(A_1 k^\beta)
\qquad\text{for all $k\ge 2$}.
\] This sort of "super intermittency" is combined with a local linearization
of the solution, and with techniques from Gaussian analysis in order to
establish the unbounded oscillations of the sample functions of the solution to
our SPDE.