On asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on real hyperbolic manifolds

Abstract

In this article, we investigate the existence, uniqueness and exponential decay of asymptotically almost periodic solutions of the parabolic-elliptic Keller-Segel system on a real hyperbolic manifold. We prove the existence and uniqueness of such solutions in the linear equation case by using the dispersive and smoothing estimates of the heat semigroup. Then we pass to the well-posedness of the semi-linear equation case by using the results of linear equation and fixed point arguments. The exponential decay is proven by using Gronwall's inequality.

Keywords:

• Keller-Segel system
• dispersive estimates
• smoothing estimates
• asymptotically almost periodic solutions
• well-posedness

2020 Mathematics Subject Classifications:

• 35A01
• 35B10
• 35B65
• 35Q30
• 35Q35
• 76D03
• 76D07

Disclosure statement

No potential conflict of interest was reported by the author(s).