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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

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Volume 9, Issue 1


Some local estimates and a uniqueness result for the entire biharmonic heat equation

Miles Simon / Glen Wheeler
  • Institut für Analysis und Numerik (IAN), Universität Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany. Current address: Institute for Mathematics and its Applications, University of Wollongong, Wollongong 2522, Australia
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Published Online: 2014-12-04 | DOI: https://doi.org/10.1515/acv-2014-0027


We consider smooth solutions to the biharmonic heat equation on ℝn × [0,T] for which the square of the Laplacian at time t is globally bounded from above by k0/t for some k0 in ℝ+, for all t ∈ [0,T]. We prove local, in space and time, estimates for such solutions. We explain how these estimates imply uniqueness of smooth solutions in this class.

Keywords: Fourth-order parabolic partial differential equations; heat flow; local estimates; uniqueness

MSC: 35B45; 35K25; 35K35

About the article

Received: 2014-08-05

Revised: 2014-10-19

Accepted: 2014-10-22

Published Online: 2014-12-04

Published in Print: 2016-01-01

Funding Source: Alexander-von-Humboldt Foundation

Award identifier / Grant number: 1137814

Funding Source: Australian Research Council Discovery Project

Award identifier / Grant number: DP120100097

Funding Source: University of Wollongong Research Council

Award identifier / Grant number: 2014 Small Grant 228381024

Citation Information: Advances in Calculus of Variations, Volume 9, Issue 1, Pages 77–99, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258, DOI: https://doi.org/10.1515/acv-2014-0027.

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