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

TL;DR

This paper investigates solutions to the Allen-Cahn equation in the Grushin plane, demonstrating that monotone solutions are stable and energy-efficient but not necessarily one-dimensional, challenging previous assumptions.

Contribution

It provides a counter-example showing monotone solutions need not be one-dimensional, despite their stability and energy properties.

Findings

Monotone solutions are stable and have good energy estimates.

Counter-example shows solutions are not always one-dimensional.

Monotonicity does not imply one-dimensionality in the Grushin plane.

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.