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Traveling wave solutions of harmonic heat flow

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Abstract

We prove the existence of a traveling wave solution of the equation \( u_t=\Delta u+|\nabla u|^2u \) in an infinitely long cylinder of radius R, which connects two locally stable and axially symmetric steady states at x 3 = ±∞. Here u is a director field with values in \( {\mathbb{S}}^2 \subset {\mathbb{R}}^3 : |u|=1 \) The traveling wave has a singular point on the cylinder axis. Letting R→ ∞ we obtain a traveling wave defined in all space.

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Authors and Affiliations

  1. Istituto per le Applicazioni del Calcolo “Mauro Picone”, CNR, V.le del Policlinico,137-00161 Rome, Italy and Dipartimento di Matematica, University of Rome Tor Vergata, Via della Ricerca Scientifica, 00133, Rome, Italy

    M. Bertsch

  2. Department of Mathematical Sciences, New Jersey Institute of Technology, Newark, NJ, 07102, USA

    C. B. Muratov

  3. Dipartimento di Matematica Guido Castelnuovo, University of Rome La Sapienza, P.le Aldo Moro, 2-00185, Rome, Italy

    I. Primi

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  1. M. Bertsch
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  3. I. Primi
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Correspondence to M. Bertsch.

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Bertsch, M., Muratov, C.B. & Primi, I. Traveling wave solutions of harmonic heat flow. Calc. Var. 26, 489–509 (2006). https://doi.org/10.1007/s00526-006-0016-2

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  • DOI: https://doi.org/10.1007/s00526-006-0016-2

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