A stochastic heat equation with non-locally Lipschitz coefficients

We consider the stochastic heat equation (SHE) on the torus T=[0,1], driven by space-time white noise W, with an initial condition u₀ that is nonnegative and not identically zero: {equation*} ∂ u/∂ t = 1/2∂^2 u/∂ x^2 + b(u) + σ(u)Ẇ. {equation*} The drift b and diffusion coefficient $σ$ 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|A₁ and σ(u)=u|log u|A₂ with A₁∈(0,1) and A₂∈(0,1/4).