Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion

We consider a simplified chemotaxis model of tumor angiogenesis, described by a Keller-Segel system on the two dimensional infinite cylindrical domain $(x, y) \in \mathbb{R} \times {\mathbf S^{\lambda}}$, where $ \mathbf S^{\lambda}$ is the circle of perimeter $\lambda>0$. The domain models a virtual channel where newly generated blood vessels toward the vascular endothelial growth factor will be located. The system is known to allow planar traveling wave solutions of an invading type. In this paper, we establish the nonlinear stability of these traveling invading waves when chemical diffusion is present if $\lambda$ is sufficiently small. The same result for the corresponding system in one-dimension was obtained by Li-Li-Wang (2014) [16]. Our result solves the problem remained open in [3] at which only linear stability of the waves was obtained under certain artificial assumption.