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This research explores how randomization affects the well-posedness of the supercritical heat equations.

May 7, 2026Journal of Dynamics and Differential Equations0 citationsOpen Access

On the effect of randomization on supercritical heat equations

Key Points

This research explores how randomization affects the well-posedness of the supercritical heat equations.
Analyzed effects of random force term on local mild solutions.
Examined non-uniqueness of solutions in the presence of additive noise.
Discussed implications of randomizing initial conditions on solution behavior.
Found non-uniqueness for local-in-time mild solutions with additive colored noise.
Demonstrated that the addition of colored noise does not improve the solution theory.
Showed that randomizing initial conditions influences well-posedness.

Abstract

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.

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