We study the singular nonlinear partial differential equation $t{\mathrm{\partial }}_{t}u=F(t,x,u,{\mathrm{\partial }}_{x}u)$, where $(t,x)\in \mathbb{R}\times {\mathbb{R}}^n$. Under some growth conditions on the coefficients of the partial Taylor expansion of $F$, we construct the unique solution that is continuous in $t$ and $C^{\mathrm{\infty }}$ in $x$.