Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain

We consider a simplified model of tumor angiogenesis, described by a Keller-Segel equation on the two dimensional domain (x,y)∈R×Sλ where Sλ is the circle of perimeter λ. It is known that the system allows planar traveling wave solutions of an invading type. In case that λ is sufficiently small, we establish the nonlinear stability of traveling wave solutions in the absence of chemical diffusion if the initial perturbation is sufficiently small in some weighted Sobolev space. When chemical diffusion is present, it can be shown that the system is linearly stable. Lastly, we prove that any solution with our front condition eventually becomes planar under certain regularity conditions.