On the effect of randomization on supercritical heat equations

Recently, in Glogić et al. (Non-uniqueness of mild solutions to supercritical heat equations. arXiv:2501.17032 (2025)), it has been shown that the focusing power nonlinearity heat equation $\begin{array}{c}{\partial }_{t}u-\Delta u={\left|u\right|}^{p-1}u,p>1,\end{array}$ in dimensions $d\ge 3$ has non-unique local solutions in $L^q\left(R^d\right)$ for $q<d(p-1)/2$ provided that $p<p_{\mathrm{JL}}$ , where $p_{\mathrm{JL}}$ denotes the Joseph-Lundgren exponent. In this paper we investigate the effect of different randomizations on the well-posedness of the equation. First we show that adding a forcing term white in time and colored in space in (NLH) is not sufficient to improve the solution theory: namely, we prove non-uniqueness for local-in-time mild solutions of (NLH) with additive noise. Second, we discuss how randomizing the initial conditions of (NLH) affects its well-posedness.