Dense blowup for parabolic SPDEs

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.