Nonnegative solutions to stochastic heat equation with nonlinear drift

We consider one-dimensional stochastic heat equation with nonlinear drift, $\displaystyle \partial_t u=\frac{1}{2}\Delta u+b(u)u+\sigma(u)\dot{W}(t,x)$, where $b:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function and $\sigma:\mathbb{R}_{+}\to \mathbb{R}$ is a continuous function with suitable property. We will construct nonnegative solutions to such SPDEs.