On the Allen-Cahn equation in theGrushin plane: A monotone entire solution that is notone-dimensional

Isabeau Birindelli; Enrico Valdinoci

doi:10.3934/dcds.2011.29.823

Article Contents

2011, Volume 29, Issue 3: 823-838. Doi: 10.3934/dcds.2011.29.823

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On the Allen-Cahn equation in the Grushin plane: A monotone entire solution that is not one-dimensional

Isabeau Birindelli^1, and
Enrico Valdinoci^2,

1.
Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro, 2, I-00185 Roma
2.
Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano

Received: November 2009

Revised: July 2010

Published: September 2011

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Abstract

We consider solutions of the Allen-Cahn equation in the whole Grushin plane and we show that if they are monotone in the vertical direction, then they are stable and they satisfy a good energy estimate.
However, they are not necessarily one-dimensional, as a counter-example shows.

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Mathematics Subject Classification: Primary: 35J15, 35J62; Secondary: 35H20, 53A35.

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