Critical norm blow-up rates for the energy supercritical nonlinear heat equation

Tobias Barker; Hideyuki Miura; Jin Takahashi

doi:10.1016/j.jfa.2026.111450

Critical norm blow-up rates for the energy supercritical nonlinear heat equation

Tobias Barker, Hideyuki Miura, Jin Takahashi

Department of Mathematical Sciences

Institute of Science Tokyo

Research output: Contribution to journal › Article › peer-review

Abstract

We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded-norm up to the blow-up time. As a consequence of this, we obtain optimal blow-up rates for certain radial solutions undergoing type II blow-up.

We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative ε-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations.

Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.

Original language	English
Article number	111450
Journal	Journal of Functional Analysis
Early online date	9 Mar 2026
DOIs	https://doi.org/10.1016/j.jfa.2026.111450
Publication status	E-pub ahead of print - 9 Mar 2026

Funding

The first author thanks Philippe Souplet for references and the Institute of Science Tokyo for its hospitality. The first author is supported by an EPSRC New Investigator Award UKRI096 ‘Dynamics and regularity criteria for nonlinear incompressible partial differential equations’. The second author is supported in part by JSPS KAKENHI Grant Numbers 17K05312, 21H00991 and 21H04433. The third author is supported in part by JSPS KAKENHI Grant Numbers 22H01131, 22KK0035 and 23K12998.

Funders	Funder number
Engineering and Physical Sciences Research Council

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title = "Critical norm blow-up rates for the energy supercritical nonlinear heat equation",

abstract = "We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded-norm up to the blow-up time. As a consequence of this, we obtain optimal blow-up rates for certain radial solutions undergoing type II blow-up. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative ε-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.",

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AU - Miura, Hideyuki

AU - Takahashi, Jin

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N2 - We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded-norm up to the blow-up time. As a consequence of this, we obtain optimal blow-up rates for certain radial solutions undergoing type II blow-up. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative ε-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.

AB - We prove the first classification of blow-up rates of the critical norm for solutions of the energy supercritical nonlinear heat equation, without any assumptions such as radial symmetry or sign conditions. Moreover, the blow-up rates we obtain are optimal, for solutions that blow-up with bounded-norm up to the blow-up time. As a consequence of this, we obtain optimal blow-up rates for certain radial solutions undergoing type II blow-up. We establish these results by proving quantitative estimates for the energy supercritical nonlinear heat equation with a robust new strategy based on quantitative ε-regularity criterion averaged over certain comparable time scales. With this in hand, we then produce the quantitative estimates using arguments inspired by Palasek [31] and Tao [38] involving quantitative Carleman inequalities applied to the Navier-Stokes equations. Our work shows that energy structure is not essential for establishing blow-up rates of the critical norm for parabolic problems with a scaling symmetry. This paves the way for establishing such critical norm blow-up rates for other nonlinear parabolic equations.

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Critical norm blow-up rates for the energy supercritical nonlinear heat equation

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