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 = 12² u x² + 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| u|^A₁ and σ (u) =u| u|^A₂ with A₁ (0, 1) and A₂ (0, 1/4).
Chen et al. (Thu,) studied this question.