[Paper Review] Kinks and solitons in linear and nonlinear-diffusion Keller-Segel type models with logarithmic sensitivity

Juan Campos, Claudia García+2 more|arXiv (Cornell University)|Feb 26, 2021

Mathematical Biology Tumor Growth29 references5 citations

TL;DR

This paper investigates traveling wave solutions—specifically kink and soliton patterns—in Keller-Segel models with logarithmic sensitivity, comparing linear and flux-saturated (relativistic) diffusion. It rigorously proves the existence of compactly supported solitons and kinks in both diffusion regimes, demonstrating that flux-saturation leads to distinct solution structures, including logarithmic convexity/concavity of the chemoattractant and infinite slope singularities at support boundaries.

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.

Motivation & Objective

• To analyze the existence of traveling wave patterns in Keller-Segel models with logarithmic chemosensitivity.
• To compare the qualitative behavior of solutions under linear diffusion versus flux-saturated (relativistic) diffusion.
• To rigorously establish the existence of solitons and kinks with compact support in both diffusion regimes.
• To characterize the structural differences in the chemoattractant and cell density profiles, particularly regarding curvature and regularity at support boundaries.
• To extend results to general flux-saturated mechanisms under integrability conditions on the inverse flux function.

Proposed method

• Formulates the Keller-Segel system with logarithmic sensitivity $ f(S) = \log S $, $ k(u,S) = u - \lambda S $, and flux function $ \Phi $ satisfying $ \Phi(-s) = -\Phi(s) $, $ \Phi' > 0 $.
• Reduces the PDE system to a traveling wave ODE system via $ u(t,x) = u(x - \sigma t) $, $ S(t,x) = S(x - \sigma t) $, introducing variables $ w = u/S $, $ v = S'/S $.
• Analyzes the phase space of the reduced system $ (w,v) $ using equilibrium point analysis and asymptotic behavior near critical points.
• Applies phase plane techniques and comparison arguments (e.g., Gronwall-type estimates) to prove existence of solutions defined on finite intervals with infinite slope at endpoints.
• Uses transformation $ w = u/S $, $ v = S'/S $ to reconstruct $ u $ and $ S $ from the $ (w,v) $ system, ensuring $ u $ has compact support and $ S $ is $ C^1 $ but not $ C^2 $ at endpoints.
• Considers two cases: (1) linear diffusion $ \Phi(s) = \mu s $, (2) relativistic flux-saturated diffusion $ \Phi(s) = \mu s / \sqrt{1 + (\mu/c)^2 s^2} $, and compares their solution structures.

Experimental results

Research questions

• RQ1 Do traveling wave solutions with compact support exist in the Keller-Segel model with logarithmic sensitivity under linear diffusion?
• RQ2 How does flux-saturation (relativistic diffusion) alter the structure and regularity of traveling wave solutions compared to linear diffusion?
• RQ3 What are the qualitative differences in the chemoattractant profile $ S $, particularly regarding logarithmic convexity/concavity and second derivative singularities?
• RQ4 Can soliton-type solutions with infinite slope at support boundaries be rigorously constructed in the flux-saturated case?
• RQ5 How do the support boundaries of $ u $ and $ S $ relate to the wave speed $ \sigma $ and parameters $ a, \lambda, \gamma, c $?

Key findings

• For linear diffusion, the paper proves the existence of solitons with compact support in $ u $, where $ u $ is zero outside a finite interval and $ S $ is logarithmically concave with $ C^1 $ but not $ C^2 $ behavior at endpoints.
• In the flux-saturated case, the paper establishes the existence of solutions with $ u $ compactly supported and $ S $ logarithmically convex, exhibiting second derivative discontinuities at $ s_{-} $ and $ s_{+} $.
• Solutions in the flux-saturated case satisfy $ \lim_{v \to \frac{\sigma \pm c}{a}} W'(v) = \mp \infty $, indicating infinite slope at the support boundaries of the $ w $-component.
• The support of $ u $ is contained in $ [s_{-}, s_{+}] = \left[\frac{\sigma - c}{a}, \frac{\sigma + c}{a}\right] $, and $ u $ is continued by zero outside this interval.
• For the flux-saturated case, the condition $ c > a\sqrt{\lambda / \gamma} $ is required to ensure existence of solutions with the described structure.
• The chemoattractant $ S $ is $ C^1 $ across the support boundary, but its second derivative has jump discontinuities, reflecting non-smoothness in the curvature.