A stochastic heat equation with non-locally Lipschitz coefficients

We consider the stochastic heat equation (SHE) on the torus $\mathbb{T}=[0,1]$, driven by space-time white noise $\dot W$, with an initial condition $u_0$ that is nonnegative and not identically zero: \begin{equation*} \frac{\partial u}{\partial t} = \tfrac{1}{2}\frac{\partial^2 u}{\partial x^2} + b(u) + \sigma(u)\dot{W}. \end{equation*} The drift $b$ and diffusion coefficient $\sigma$ are Lipschitz continuous away from zero, although their Lipschitz constants may blow up as the argument approaches zero. We establish the existence of a unique global mild solution that remains strictly positive. Examples include $b(u)=u|\log u|^{A_1}$ and $\sigma(u)=u|\log u|^{A_2}$ with $A_1\in(0,1)$ and $A_2\in(0,1/4)$.