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15 April 2021 Optimal Liouville theorems for superlinear parabolic problems
Pavol Quittner
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Duke Math. J. 170(6): 1113-1136 (15 April 2021). DOI: 10.1215/00127094-2020-0096

Abstract

Liouville theorems for scaling invariant nonlinear parabolic equations and systems (saying that the equation or system does not possess positive entire solutions) guarantee optimal universal estimates of solutions of related initial and initial-boundary value problems. In the case of the nonlinear heat equation

utΔu=upinRn×R,p>1,

the nonexistence of positive classical solutions in the subcritical range p(n2)<n+2 has been conjectured for a long time, but all known results require either a more restrictive assumption on p or deal with a special class of solutions (time-independent or radially symmetric or satisfying suitable decay conditions). We solve this open problem and—by using the same arguments—we also prove optimal Liouville theorems for a class of superlinear parabolic systems. In the case of the nonlinear heat equation, straightforward applications of our Liouville theorem solve several related long-standing problems. For example, they guarantee an optimal Liouville theorem for ancient solutions, optimal decay estimates for global solutions of the corresponding Cauchy problem, optimal blowup rate estimate for solutions in nonconvex domains, and optimal universal estimates for solutions of the corresponding initial-boundary value problems. The proof of our main result is based on refined energy estimates for suitably rescaled solutions.

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Pavol Quittner. "Optimal Liouville theorems for superlinear parabolic problems." Duke Math. J. 170 (6) 1113 - 1136, 15 April 2021. https://doi.org/10.1215/00127094-2020-0096

Information

Received: 29 March 2020; Revised: 26 August 2020; Published: 15 April 2021
First available in Project Euclid: 23 March 2021

Digital Object Identifier: 10.1215/00127094-2020-0096

Subjects:
Primary: 35B40
Secondary: 35B44 , 35B45 , 35K58 , 35K61

Keywords: Liouville theorems , superlinear parabolic problems

Rights: Copyright © 2021 Duke University Press

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Vol.170 • No. 6 • 15 April 2021
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