Heteroclinic traveling waves of 1D parabolic systems with degenerate stable states

TL;DR

This paper proves the existence of heteroclinic traveling waves in 1D parabolic systems with degenerate stable states, extending prior results by removing non-degeneracy assumptions on minima.

Contribution

It introduces a new approach that allows for potentials with degenerate minima, broadening the class of systems where heteroclinic traveling waves are known to exist.

Findings

Existence of heteroclinic traveling waves for degenerate minima.

Main results apply to curves in general Hilbert spaces.

Extension of previous work to more general potential functions.

Abstract

We study the existence of traveling waves for the parabolic system \begin{equation} \partial_t w - \partial_{x}^2 w = -\nabla_{\mathbb{u}} W(w) \mbox{ in } [0,+\infty) \times \mathbb{R} \end{equation} where $W$ is a potential bounded below and possessing two minima at different levels. We say that $\mathfrak{w}$ is a traveling wave solution of the previous equation if there exist $c^\star>0$ and $\mathfrak{u} \in \mathcal{C}^2(\mathbb{R},\mathbb{R}^k)$ such that $\mathfrak{w}(t,x)=\mathfrak{u}(x-ct)$. For a class of potentials $W$, heteroclinic traveling waves of the previous equation where shown to exist by Alikakos and Katzourakis \cite{alikakos-katzourakis}. More precisely, assuming the existence of two local minimizers of $W$ at \textit{different} levels which, in addition, satisfy some non-degeneracy assumptions, the authors in \cite{alikakos-katzourakis} show the existence of a speed $c^\star>0$ and profile $\mathfrak{u} \in \mathcal{C}^2(\mathbb{R},\mathbb{R}^k)$ such that $\mathfrak{u}$ connects the two local minimizers at infinity. In this paper, we show that the non-degeneracy assumption on the local minima can be dropped and replaced by another one which allows for potentials possessing degenerate minima. As we do in \cite{oliver-bonafoux-tw}, our main result is in fact proven for curves which take values in a general Hilbert space and the main result is deduced as a particular case, in the spirit of the earlier works by Monteil and Santambrogio \cite{monteil-santambrogio} and Smyrnelis \cite{smyrnelis} devoted to the existence of stationary heteroclinics.