Asymptotic behaviors and existence of traveling wave solutions to the Keller-Segel model with logarithmic sensitivity

Abstract

<p style='text-indent:20px;'>In this paper, we are concerned with the existence of traveling wave solutions for the Keller-Segel model with logarithmic sensitivity. By the Hopf-Cole transformation and traveling wave transformation, the degenerate Keller-Segel system is transformed into a singularly perturbed system. By constructing an invariant region to prove the existence of the traveling wave solutions for the degenerate system, we obtain the traveling wave solutions for Keller-Segel system with small parameter by using geometric singular perturbation theory and Fredholm theory. Finally, we discuss the asymptotic behaviors of the traveling wave solutions.</p>

A geometric construction of travelling waves to Keller-Segel type diffusion system with local or nonlocal delay

“…Compared with the previous works [5,11,14], our study provides a comprehensive establishment of the asymptotic behaviors of travelling waves. Similar to the work in [23], both the existence and asymptotic behaviors of travelling waves for the KS type diffusion system were both established. However, unlike [23], the invariant region and the asymptotic behaviors constructed in this paper are more complex and exhibit higher dimensionality.…”

“…Remark 5.4. When compared with works in [2,5,11,14,23,25], the enhancement of time delay presents significant challenges in exploring its complex effects on the dynamics of the system. To the best of the authors' knowledge, there are very few studies that focus on the role of time delay in the KS system.…”