Spectrum of the M 5 -traveling waves

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 (ℝ; 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 w 2 (ℝ; 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 wα 2 (ℝ; ℂ 2 ) × L wε 2 (ℝ; ℂ) the essential spectrum lies on { Reλ