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

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.