Traveling waves and their spectral instability in a time-fractional chemotactic model

In this paper, one considers a time-fractional chemotactic model with the small cell diffusion coefficient and chemotactic sensitivity. By the geometric singular perturbation theory together with the stable manifold theorem, one gets the existence of the positive traveling waves connecting the two constant steady states $ (0,0) $ and $ (1,\frac{a}{b}) $ with the regular case $ c = O(1) $ and slow case $ c = O(\sqrt{\epsilon}) $, respectively. In addition, by the spectral analysis it shows that the above traveling waves are spectrally unstable in the Banach space $ C_{ub}(\mathbb{R},\mathbb{R}^2) $ formed by the bounded and uniformly continuous functions.