This article revisits the problem of global well-posedness for the generalized parabolic Anderson model on \(\mathbb {R}^+\times \mathbb {T}^2\) within the framework of paracontrolled calculus (Gubinelli et al. in Forum Math, 2015). The model is given by the equation: \(\begin{aligned} (\partial _t-\Delta ) u=F(u)\eta \end{aligned}\) where \(\eta \in C^{-1-\kappa }\) with \(1/6>\kappa >0\) , and \(F\in C_b^2(\mathbb {R})\) . Assume that \(\eta \in C^{-1-\kappa }\) and can be lifted to enhanced noise, we derive new a priori bounds. The key idea follows from the recent work by Chandra et al. (A priori bounds for 2-d generalised Parabolic Anderson Model, , 2024), to represent the leading error term as a transport type term, and our techniques encompass the paracontrolled calculus, the maximum principle, and the localization approach (i.e. high-low frequency argument).
