Kinks and solitons in linear and nonlinear-diffusion Keller-Segel type models with logarithmic sensitivity

TL;DR

The paper advances the understanding of traveling wave patterns in Keller–Segel systems with logarithmic sensitivity by contrasting linear diffusion with flux-saturated diffusion, such as the relativistic heat equation. It derives a traveling-wave ODE in terms of $w=u/S$ and $v=S'/S$, enabling a detailed phase-plane analysis. In the linear-diffusion regime, it proves the existence of block-type and finite-mass traveling waves, structured by separatrix curves $\Gamma(P)$ and governed by parameter regimes tied to diffusion and chemotactic strength. In the flux-saturated regime, it uncovers novel umbrella-type waves and shows that a continuous separating curve $\Lambda$ delineates initial data yielding traveling waves, highlighting qualitative differences induced by flux limitation. Overall, the work provides rigorous classifications of traveling wave profiles and thresholds for their existence, enriching the theory of chemotaxis models with logarithmic sensitivity and flux-limited diffusion.

Abstract

This paper investigates the existence of traveling--wave--type patterns in the Keller--Segel model with logarithmic sensitivity. We consider both the linear diffusion case and the nonlinear, flux-saturated diffusion of relativistic heat--equation type, providing a detailed comparison between the two regimes. Particular attention is devoted to traveling waves exhibiting compact support or support restricted to a half-line. We rigorously establish the existence of such patterns and highlight the qualitative differences arising from the choice of diffusion mechanism.

Figures (7)

• Figure 1: Figures A, B and C correspond to the linear diffusion, in the cases $0<a<1$, $a=1$ and $a>1$ respectively. Figures $\textnormal{A}^*$, $\textnormal{B}^*$, and $\textnormal{C}^*$ correspond to the same cases and complete the set of finite–mass solutions for the problem with linear diffusion. .Figures D and E correspond to the flux--saturated mechanisms case.
• Figure 2: Qualitative illustration of Theorem \ref{['teorema1']}. Panels A, B, and C display the phase portraits of the dynamical system for $\sigma=0$, while Panel D corresponds to $\sigma=\sqrt{\frac{\lambda\gamma(1-a)^2}{\gamma-\delta(1-a)}}$. Red curves denote the isoclines of the system, and blue curves represent the trajectories $\Gamma$. The grey--shaded region indicates the parameter zone in which the solution exhibits semi-block behaviour at $\sigma=\sigma^*$, whereas the horizontally shaded region marks the set where no finite mass solution exists for $\sigma=\sigma^*$.
• Figure 3: Qualitative representation of Theorem \ref{['teorema2']}
• Figure 4: Top panels (A-B): case $a<1$. Middle panel (C): case $a=1$ (or formally $\sigma=\infty$). Bottom panels (D-E): case $a>1$. Left column (A, D): regime $P_3\in\Omega$. Right column (B, E): regime $P_3\notin\Omega$. Red curves represent the isoclines of the system, and the black arrows illustrate the corresponding flow directions. The blue lines represent the stable and unstable vector of the equilibrium points described in Proposition \ref{['prophiperbolic']}.
• Figure 5: Representación de los conjuntos $B_1$ y $B_2$ de la demostración de la Proposición \ref{['propP3Hiper']}.

Theorems & Definitions (50)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Proposition 2.1
• proof
• Lemma 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.3