Related papers: Quantitative analysis and its applications for Kel…
The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term $-\mu \rho^2$ (modeling…
This paper studies the controllability for a Keller-Segel type chemotaxis model with singular sensitivity. Based on the Hopf-Cole transformation, a nonlinear parabolic system, which has first-order couplings, and the coupling coefficients…
This paper investigates the Keller-Segel model with quadratic cellular diffusion over a disk in $\mathbb R^2$ with a focus on the formation of its nontrivial patterns. We obtain explicit formulas of radially symmetric stationary solutions…
We consider the Keller-Segel system with logical source \begin{align*} \begin{cases} u_t = \nabla \cdot (\phi(u)\nabla u) - \nabla \cdot (\psi(u)\nabla v)+f(u), & x \in \Omega, \; t > 0, v_t = \Delta v - v + u, & x \in \Omega, \; t > 0,…
We present a generalized Keller-Segel model where an arbitrary number of chemical compounds react, some of which are produced by a species, and one of which is a chemoattractant for the species. To investigate the stability of homogeneous…
This paper investigates an initial-Neumann boundary value problem for a Keller--Segel system with parabolic-parabolic-ODE coupling. The model incorporates a signal-dependent, non-increasing motility function that, through indirect signal…
We study the finite-time blow-up in two variants of the parabolic-elliptic Keller-Segel system with nonlinear diffusion and logistic source. In $n$-dimensional balls, we consider \begin{align*} \begin{cases} u_t = \nabla \cdot…
In this paper we study the controllability of a coupled Keller-Segel-Navier-Stokes system. We show the local exact controllability of the system around some particular trajectories. The proof relies on new Carleman inequalities for the…
We consider an initial-boundary value problem for the chemotaxis-Navier--Stokes system \begin{align*} \left\{ \begin{array}{c@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\nabla n=\nabla\cdot\big(D(n)\nabla n-nS(x,n,c)\cdot\nabla c\big),\…
We consider a coupled system of Keller-Segel type equations and the incompressible Navier-Stokes equations in spatial dimension two and three. In the previous work [19], we established the existence of a weak solution of a Fokker-Plank…
A finite volume scheme for the (Patlak-) Keller-Segel model in two space dimensions with an additional cross-diffusion term in the elliptic equation for the chemical signal is analyzed. The main feature of the model is that there exists a…
In this paper, the fully parabolic Keller-Segel system \begin{equation} \left\{ \begin{array}{llc} u_t=\Delta u-\nabla\cdot(u\nabla v), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u, &(x,t)\in\Omega\times (0,T),\\ \end{array} \right.…
Chemotaxis is a fundamental mechanism of cells and organisms, which is responsible for attracting microbes to food, embryonic cells into developing tissues, or immune cells to infection sites. Mathematically chemotaxis is described by the…
Solutions $(u, v)$ to the chemotaxis system \begin{align*} \begin{cases} u_t = \nabla \cdot ( (u+1)^{m-1} \nabla u - u (u+1)^{q-1} \nabla v), \\ \tau v_t = \Delta v - v + u \end{cases} \end{align*} in a ball $\Omega \subset \mathbb R^n$, $n…
We use De Giorgi-Nash-Moser iteration scheme to establish that weak solutions to a coupled system of elliptic equations with critical growth on the boundary are in $L^\infty(\Omega)$. Moreover, we provide an explicit $L^\infty(\Omega)$-…
We consider two-dimensional versions of the Keller--Segel model for the chemotaxis with either classical (Brownian) or fractional (anomalous) diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived.…
We consider the following chemotaxis model %fully parabolic Keller-Segel system with logistic source $$ \left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)+\mu (u-u^2),\quad x\in \Omega, t>0, \disp{v_t-\Delta…
We perform a Lie symmetry analysis on the tempered-fractional Keller Segel (TFKS) system, a chemo-taxis model incorporating anomalous diffusion. A novel approach is used to handle the nonlocal nature of tempered fractional operators. By…
This paper investigates the following Keller-Segel-Navier-Stokes system with nonlinear diffusion and rotational flux $$\begin{align}\begin{cases} &n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(nS(x, n, c)\nabla c),\quad &x\in \Omega, t>0, \\…
The purpose of this work is the study of \textit{chemotaxis} and how to model it through the equations of Keller-Segel. \textit{Chemotaxis} is a natural process which induces the organisms to direct their movement according to certain…