Can the Stochastic Wave Equation with Strong Drift Hit Zero?

We study the stochastic wave equation with multiplicative noise and singular drift: $$\partial_tu(t,x)=\Delta u(t,x)+u^{-\alpha}(t,x)+g(u(t,x))\dot{W}(t,x)$$ where $x$ lies in the circle $\mathbf{R}/J\mathbf{Z}$ and $u(0,x)>0$. We show that (i) If $0<\alpha<1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$. (ii) If $\alpha>3$ then with probability one, $u(t,x)\ne0$ for all $(t,x)$.