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

Pages: 823 - 838, Volume 29, Issue 3, March 2011

doi:10.3934/dcds.2011.29.823       Abstract        References        Full Text (412.3K)       Related Articles

Isabeau Birindelli - Dipartimento di Matematica, Università di Roma La Sapienza, Piazzale Aldo Moro, 2, I-00185 Roma, Italy (email)
Enrico Valdinoci - Università degli Studi di Milano, Dipartimento di Matematica Via Saldini, 50, 20133 Milano, Italy (email)

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.

Keywords:  Symmetry problmes, Allen-Cahn equation, Grushin plane.
Mathematics Subject Classification:  Primary: 35J15, 35J62; Secondary: 35H20, 53A35.

Received: November 2009;      Revised: July 2010;      Published: November 2010.

 References