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.
On the allen-cahn equation in the grushin plane: a monotone entire solution that is not one-dimensional / I. Birindelli, E. Valdinoci. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - 29:3(2011 Mar), pp. 823-838.
On the allen-cahn equation in the grushin plane: a monotone entire solution that is not one-dimensional
E. ValdinociUltimo
2011
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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