PUB - Publikationen an der Universität Bielefeld

Recently, in Glogic 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 partial derivative(t)u-Delta u = |u|(p-1)u, p > 1, (NLH) in dimensions d >= 3 has non-unique local solutions in L-q(R-d) for q < d(p-1)/2 provided that p < p(JL), where 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.