Emergence of large densities in a chemotaxis system with signal-dependent motility
Abstract
In this paper, we focus on the emergence of large densities of a cell population in finite time in the following parabolic-elliptic Keller-Segel system with signal-dependent motility and a source term of logistic type uε t = Δ(e−vεuε) + ε(κuε − μuε2) , 0 =Δvε−vε+ uε. in a smooth bounded domain Ω⊆ℝ2 with homogenuous Neumann boundary conditions and parameters κ,μ>0 and ε∈(0,1). We will show that the solution uε can exceed any certain value M>0 before a time T(M)>0 under the condition that ε>0 is sufficiently small. To prove the result, we show convergence of classical solutions of the first system to classical solutions of the corresponding limit system as εsearrow 0 ut=Δ(e−vu), −Δv+v=u. The main challenge is to find suitable bounds for uε and vε, that are ε-independent, in Lebesgue and Hölder spaces, whereas for fixed ε>0 global existence and boundedness is known.
Related
- Global existence and uniqueness of solutions in a chemotaxis system with density-suppressed motility, a θ−logistic source, and indirect signal production under a strong Allee effect
- Nonlinear Stability of Nonconstant Steady States in Chemotaxis--Consumption Systems with General Motility
- Global Boundedness and Finite-Time Blow-up for a Quasilinear Chemotaxis--May--Nowak System
- Local null controllability of a parabolic-elliptic chemotaxis system by a Neumann boundary control
- Large-time behavior in an attraction--repulsion chemotaxis system
BibTeX
@misc{hegenbart2026emergence,
title = {{Emergence of large densities in a chemotaxis system with signal-dependent motility}},
author = {Max Hegenbart},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.36934},
eprint = {2610.36934},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.36934},
}
RIS
TY - UNPB
TI - Emergence of large densities in a chemotaxis system with signal-dependent motility
AU - Max Hegenbart
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.36934
UR - https://arxiv.org/abs/2610.36934
AB - In this paper, we focus on the emergence of large densities of a cell population in finite time in the following parabolic-elliptic Keller-Segel system with signal-dependent motility and a source term of logistic type u_ε t = Δ(e^−v_εu_ε) + ε(κu_ε − μu_ε^2) , 0 =Δv_ε−v_ε+ u_ε. in a smooth bounded domain Ω⊆ℝ^2 with homogenuous Neumann boundary conditions and parameters κ,μ>0 and ε∈(0,1). We will show that the solution u_ε can exceed any certain value M>0 before a time T(M)>0 under the condition that ε>0 is sufficiently small. To prove the result, we show convergence of classical solutions of the first system to classical solutions of the corresponding limit system as εsearrow 0 u_t=Δ(e^−vu), −Δv+v=u. The main challenge is to find suitable bounds for u_ε and v_ε, that are ε-independent, in Lebesgue and Hölder spaces, whereas for fixed ε>0 global existence and boundedness is known.
ER -