Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions

By Michael Winkler. Advanced Nonlinear Studies, 2020

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Abstract The chemotaxis-growth system ($\star$) { u t = D ⁢ Δ ⁢ u - χ ⁢ ∇ ⋅ ( u ⁢ ∇ ⁡ v ) + ρ ⁢ u - μ ⁢ u α , v t = d ⁢ Δ ⁢ v - κ ⁢ v + λ ⁢ u {}\left\{\begin{aligned} \displaystyle{}u_{t}&\displaystyle=D\Delta u-\chi% \nabla\cdot(u\nabla v)+\rho u-\mu u^{\alpha},\\ \displaystyle v_{t}&\displaystyle=d\Delta v-\kappa v+\lambda u\end{aligned}\right. is considered under homogeneous Neumann boundary conditions in smoothly bounded domains Ω⊂ℝn{\Omega\subset\mathbb{R}^{n}}, n≥1{n\geq 1}. For any choice of α>1{\alpha>1}, the literature provides a comprehensive result on global existence for widely arbitrary initial data within a suitably generalized solution concept, but the regularity properties of such solutions may be rather poor, as indicated by precedent results on the occurrence of finite-time blow-up in corresponding parabolic-elliptic simplifications. Based on the analysis of a certain eventual Lyapunov-type feature of ($\star$), the present work shows that, whenever α≥2-2n{\alpha\geq 2-\frac{2}{n}}, under an appropriate smallness assumption on χ, any such solution at least asymptotically exhibits relaxation by approaching the nontrivial spatially homogeneous steady state ((ρμ)1α-1,λκ⁢(ρμ)1α-1){\bigl{(}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}},\frac{\lambda}{% \kappa}\bigl{(}\frac{\rho}{\mu}\bigr{)}^{\frac{1}{\alpha-1}}\bigr{)}} in the large time limit.

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/e8e45d8031f187218bc3362677e8e8e3655e1932

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Citations in the corpus

Logistic damping effect in chemotaxis models with density-suppressed motility (2021, background): “Since the proof has some differences from [42], we provide the details for clarity.”