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We consider a simplified chemotaxis model of tumor angiogenesis, described bya 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 channelwhere newly generated blood vessels toward the vascular endothelial growthfactor will be located. The system is known to allow planar traveling wavesolutions of an invading type. In this paper, we establish the nonlinearstability of these traveling invading waves when chemical diffusion is presentif $\lambda$ is sufficiently small. The same result for the correspondingsystem in one-dimension was obtained by Li-Li-Wang (2014) [16]. Our resultsolves the problem remained open in [3] at which only linear stability of thewaves was obtained under certain artificial assumption.

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