Global solution for superlinear stochastic heat equation on ${\mathbb{R}}^d$ under Osgood-type conditions

We study the $\mathit{\text{stochastic}}\mathit{\text{heat}}\mathit{\text{equation}}$ (SHE) on ${\mathbb{R}}^d$ subject to a centered Gaussian noise that is white in time and colored in space.The drift term is assumed to satisfy an Osgood-type condition and the diffusion coefficient may have certain related growth. We show that there exists random field solution which do not explode in finite time. This complements and improves upon recent results on blow-up of solutions to stochastic partial differential equations.