Finite-Time Transition to Intermittency for a Stochastic Heat Equation Driven by the Square of a Gaussian Field

In this paper, we study the spatial behavior of the solution $ψ(x,t)$ to the stochastic heat equation $\partial_tψ(x,t)-\frac{1}{2}\partial^2_{x^2} ψ(x,t)=g\, S(x,t)^2\, ψ(x,t)$, with $0\le t\le T$, $x\in\mathbb{R}$, and $ψ(x,0)=1$. Here, $g>0$ is a coupling constant and $S(x,t)$ is a stationary, ho