Solvability of a Keller-Segel system with signal-dependent sensitivity\n and essentially sublinear production

2018/07/26 by Giuseppe Viglialoro, Thomas E. Woolley, Viglialoro, Giuseppe +1 · 1 citation
Biochemistry, Genetics and Molecular Biology · Mathematics · Medicine · #Analysis of PDEs (math.AP) #Cellular Mechanics and Interactions #FOS: Mathematics #Gene Regulatory Network Analysis #Mathematical Biology Tumor Growth #Mathematical and Theoretical Epidemiology and Ecology Models

paper · pdf · doi:10.48550/arxiv.1807.10005

openalex publication_date 2018/07/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper we consider the zero-flux chemotaxis-system n
begincases u_t=
Delta u-
nabla
cdot (u
chi(v)
nabla v) amp;\n
textrmin
quad
Omega
times (0,
infty),

0=
Delta v-v+g(u) amp;\n
textrmin
quad
Omega
times (0,
infty),

in a smooth and\nbounded domain \Ω of \ℝ². The chemotactic sensitivity \χ\nis a general nonnegative function from C¹((0,\∞)) whilst g, the\nproduction of the chemical signal v, belongs to C¹([0,\∞)) and\nsatisfies \λ₁\≤ g(s)\≤ \λ₂(1+s)^\β, for all s\≥ 0,\n0\≤\β\≤ \(1)/(2) and 0<\λ₁\≤ \λ₂. It is established\nthat no chemotactic collapse for the cell distribution u occurs in the sense\nthat any arbitrary nonnegative and sufficiently regular initial data u(x,0)\nemanates a unique pair of global and uniformly bounded functions (u,v) which\nclassically solve the corresponding initial-boundary value problem. Finally, we\nillustrate the range of dynamics present within the chemotaxis system by means\nof numerical simulations.\n

Citations

• On explosions of solutions to a system of partial differential equations modelling chemotaxis
• A user’s guide to PDE models for chemotaxis
• Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model

Cited by

• Global existence and stability in a class of chemotaxis systems with lethal interactions, nonlinear diffusion and production

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