  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.
