Abstract 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 aligned ₜ u - u = |u|^p-1u, p>1, aligned ∂ t u - Δ u = | u | p - 1 u, p > 1, in dimensions d 3 d ≥ 3 has non-unique local solutions in Lq (Rᵈ) L q (R d) for q q d (p - 1) / 2 provided that p p p JL, where p₉₋ p 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.
Eliseo Luongo (Wed,) studied this question.