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.
Bibliographical noteFunding Information:
We thank the anonymous reviewer for careful reading and helpful comments that improve the quality of our manuscript. MC's work was supported by NRF - 2015R1C1A2A01054919 . KC's work was supported by the 2015 Research Fund ( 1.150064.01 ) of UNIST (Ulsan National Institute of Science and Technology) and by NRF - 2015R1D1A1A01058614 . KK's work was partially supported by NRF - 2017R1A2B4006484 and NRF - 20151009350 . JL's work was supported by NRF - 2009-0083521 .
All Science Journal Classification (ASJC) codes
- Applied Mathematics