This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains

/pdf/locally-bounded-global-solutions-in-a-three-dimensional-3w3lqjufj4.pdf

Locally bounded global solutions in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion

http://jxshix.people.wm.edu/shi/2016-Wu-Shi-Wu-JDE.pdf

Global existence of solutions and uniform persistence of a diffusive predator-prey model with prey-taxis ✩

The coupled chemotaxis–fluid system

 $$\left\{ \begin{array}{lll} &n_t + u\cdot \nabla n = \Delta n - \nabla \cdot (n \nabla c) +rn-\mu n^2, \\ & c_t + u\cdot \nabla c = \Delta c-c+n , \\ & u_t + \nabla P = \Delta u + n \nabla \phi + g(x,t), \\ & \nabla \cdot u = 0, \end{array}\right.$$
 is considered under no-flux boundary conditions for n and c and no-slip boundary conditions for u in three-dimensional bounded domains with smooth boundary, where $${r\geq 0}$$
 and $${\mu > 0}$$
 are given constants and $${\phi\in W^{1, \infty}(\Omega)}$$
 and $${g\in C^1(\bar\Omega\times [0, \infty)) \cap L^\infty(\Omega\times (0,\infty))}$$
 are prescribed parameter functions. It is shown that under the explicit condition $${\mu\geq 23}$$
 and suitable regularity assumptions on the initial data, the corresponding initial-boundary problem possesses a global classical solution which is bounded. Apart from this, it is proved that if r = 0, then both n(·, t) and c(·, t) decay to zero with respect to the norm in $${L^\infty(\Omega)}$$
 as $${t\to \infty}$$
 , and that if, moreover, $${\int_0^\infty \int_\Omega |g|^2 < \infty}$$
 , then also u(·, t)→ 0 in $${L^\infty(\Omega)}$$
 as $${t\to \infty}$$
 .

Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis–fluid system

This paper deals with the chemotaxis system
\[
	\begin{cases}
	u_t=\Delta u - \nabla \cdot (u\nabla v),
	\qquad x\in \Omega, \ t>0, \\
	v_t=\Delta v + wz,
	\qquad x\in \Omega, \ t>0, \\
	w_t=-wz,
	\qquad x\in \Omega, \ t>0, \\
	z_t=\Delta z - z + u,
 \qquad x\in \Omega, \ t>0,
	\end{cases}
\]
in a smoothly bounded domain $\Omega \subset \mathbb{R}^n$, $n \le 3$,
that has recently been proposed as a model for tumor invasion
in which the role of an active extracellular matrix is accounted for.
 
It is shown that for any choice of nonnegative and suitably regular initial data $(u_0,v_0,w_0,z_0)$, a corresponding
initial-boundary value problem of Neumann type possesses a global solution which is bounded.
Moreover, it is proved that whenever $u_0\not\equiv 0$, these solutions approach a certain
spatially homogeneous equilibrium in the sense that as $t\to\infty$,
 
	$u(x,t)\to \overline{u_0}$ ,  &nbsp
	$v(x,t) \to \overline{v_0} + \overline{w_0}$,  &nbsp
	$w(x,t) \to 0$  &nbsp and &nbsp $z(x,t) \to \overline{u_0}$, &nbsp
uniformly with respect to $x\in\Omega$, where $\overline{u_0}:=\frac{1}{|\Omega|} \int_{\Omega} u_0$,
$\overline{v_0}:=\frac{1}{|\Omega|} \int_{\Omega} v_0$ &nbspand  &nbsp $\overline{w_0}:=\frac{1}{|\Omega|} \int_{\Omega} w_0$.

Stabilization in a chemotaxis model for tumor invasion

Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier–Stokes system with competitive kinetics

Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type

Global Existence and Smoothing Effect for the Complex Ginzburg-Landau Equation with p-Laplacian

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

Tomomi Yokota

Author Tools

Chat about Author