Spectrum of the M⁵-traveling waves

Abstract

In this paper we study the essential spectrum of the operator obtained by linearizing at traveling waves that occur in the one-dimensional version of the M$^5$-model for mesenchymal cell movement inside a directed tissue made up of highly aligned fibers. We show that traveling waves are spectrally unstable in $L^2(\mathbb{R};\mathbb{C}^3)$ as the essential spectrum includes the imaginary axis. Tools in the proof include exponential dichotomies and Fredholm properties. We prove that a weighted space $L^2_{w}(\mathbb{R};\mathbb{C}^3)$ with the same function for the tree variables of the linearized operator is no suitable to shift the essential spectrum to the left of the imaginary axis. We find a pair of appropriate weight functions whereby on the weighted space $L^2_{w_{\alpha}}(\mathbb{R};\mathbb{C}^2)\times L^2_{w_{\varepsilon}}(\mathbb{R};\mathbb{C})$ the essential spectrum lies on $\left\{\mathfrak{R}e\hspace{.03cm}\lambda<0\right\}$, outside the imaginary axis.