Nonnegative solutions to stochastic heat equation with nonlinear drift

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

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