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

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


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