A quasilinear bistable equation in cylinders and timelike heteroclinics in special relativity

TL;DR

This paper studies a quasilinear bistable equation in cylindrical domains within the context of special relativity, establishing existence, uniqueness, and one-dimensionality of phase transition solutions and exploring conditions for heteroclinic connections.

Contribution

It proves the existence, uniqueness, and one-dimensionality of minimizers for a relativistic bistable equation and investigates conditions for heteroclinic connections in non-autonomous models.

Findings

Existence of smooth minimizers for the relativistic phase transition model.

Uniqueness of solutions up to translation.

Conditions on the coefficient function for heteroclinic connection existence.

Abstract

In this note we consider the action functional \[ \int_{\mathbb{R} \times \omega} \left( 1 - \sqrt{ 1 - |\nabla u|^2 } + W(u) \right) \, \mathrm{d}t, \] where $W$ is a double well potential and $\omega$ is a bounded domain of $\mathbb{R}^{N-1}$. We prove existence, one-dimensionality and uniqueness (up to translation) of a smooth minimizing phase transition between the two stable states $u=1$ and $u=-1$. The question of existence of at least one minimal heteroclinic connection for the non autonomous model \[ \int_{\mathbb{R}} \left( 1 - \sqrt{1-|u'|^2} + a(t) W(u) \right) \, \mathrm{d}t \] is also addressed. For this, we look for the possible assumptions on $a(t)$ ensuring the existence of a minimizer.