Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity

By Xinru Cao. 2017

65 citations recorded by the source corpus.

The fully parabolic Keller-Segel system with logistic source \begin{document}$\begin{equation} \left\{ \begin{array}{llc} \displaystyle u_t=\Delta u-\chi\nabla\cdot(u\nabla v)+\kappa u-\mu u^2, &(x,t)\in \Omega\times (0,T),\\ \displaystyle \tau v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T), \end{array} \right.(\star) \end{equation}$\end{document} is considered in a bounded domain $\Omega\subset\mathbb{R}^N$ ($N≥ 1$) under Neumann boundary conditions, where $κ∈\mathbb{R}$, $μ>0$, $χ>0$ and $τ>0$. It is shown that if the ratio $\frac{χ}{μ}$ is sufficiently small, then any global classical solution $(u, v)$ converges to the spatially homogenous steady state $(\frac{κ_+}{μ}, \frac{κ_+}{μ})$ in the large time limit. Here we use an approach based on maximal Sobolev regularity and thus remove the restrictions $τ=1$ and the convexity of $\Omega$ required in [ 17 ].

Subfields: Cell Biology, Modeling and Simulation, Molecular Biology

View Paper: https://www.semanticscholar.org/paper/ea3f9948f74e749f06f4e293fd7e1a2f6e6106d0

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Cell Biology

In Modeling and Simulation

In Molecular Biology

Citations in the corpus

Attractiveness of Constant States in Logistic-Type Keller–Segel Systems Involving Subquadratic Growth Restrictions (2020, background): “In the best understood case α = 2 of quadratic absorption, for instance, associated no-flux initial-boundary value problems in n-dimensional bounded domainsΩ are known to admit global bounded classical solutions for all suitably regular initial data if either n ≤ 2 (see [26, 27]), or n ≥ 3 and μ > μ”