Related papers: Critical mass for infinite-time aggregation in a c…

We study the Neumann initial-boundary problem for the chemotaxis system \begin{align*} \left\{\begin{array}{c@{\,}l@{\quad}l@{\,}c} u_{t}&=\Delta u-\nabla\!\cdot(u\nabla v),\ &x\in\Omega,& t>0,\\ v_{t}&=\Delta v-v+u+f(x,t),\ &x\in\Omega,&…

The Neumann initial-boundary problem for the chemotaxis system \begin{align} \label{prob:abstract} \tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \kappa(|x|) u - \mu(|x|) u^p, \\ 0 = \Delta v -…

We analyze blowup solutions in infinite time of the Neumann boundary value problem for the fully parabolic chemotaxis system with local sensing: \begin{equation*} \begin{cases} u_t = \Delta(e^{-v}u)\qquad &\mathrm{in}\ \Omega \times…

The Neumann problem in balls $\Omega\subset\mathbb{R}^n$, $n\in\{3,4\}$, for the chemotaxis system \begin{equation*} \left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - \mu^{(w)}(t) + w, \quad…

We study the Neumann initial-boundary value problem for the parabolic-elliptic chemotaxis system, proposed by J\"ager and Luckhaus (1992). We confirm that their comparison methods can be simplified and refined, applicable to seek the…

In this paper, we deal with quasilinear Keller--Segel systems with indirect signal production, $$\begin{cases} u_t = \nabla \cdot ((u+1)^{m-1}\nabla u) - \nabla \cdot (u \nabla v), &x \in \Omega,\ t> 0,\\ 0 = \Delta v - \mu(t) + w, &x \in…

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system \begin{equation}\tag{$\ast$} \label{ks0} \left\{ \begin{aligned} u_t =&\; \Delta u - \nabla \cdot(u \nabla v) \quad in {\mathbb R}^2\times(0,\infty),\\ v…

This paper is concerned with a parabolic-parabolic-parabolic chemotaxis system with indirect signal production, modelling the impact of phenotypic heterogeneity on population aggregation \begin{equation*} \begin{cases} u_t = \Delta u -…

This paper is concerned with a three-component chemotaxis model accounting for indirect signal production,reading as $u_t = \nabla\cdot(\nabla u - u\nabla v)$,$v_t = \Delta v - v + w$ and $0 = \Delta w - w + u$,posed in a ball of $\mathbb…

Our main result shows that the mass $2\pi$ is critical for the minimal Keller-Segel system \begin{align}\label{prob:abstract}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - v + u, \end{cases}…

Derived from a biophysical model for the motion of a crawling cell, the system \[(*)~\begin{cases}u_t=\Delta u-\nabla\cdot(u\nabla v)\\0=\Delta v-kv+u\end{cases}\] is investigated in a finite domain $\Omega\subset\mathbb{R}^n$, $n\geq2$,…

The goal of this paper is to exhibit a critical mass phenomenon occuring in a model for cell self-organization via chemotaxis. The very well known dichotomy arising in the behavior of the macroscopic Keller-Segel system is derived at the…

We consider a parabolic-elliptic Keller-Segel system with spatially dependent diffusion sensitivity \begin{eqnarray*} \left\{ \begin{array}{l} u_t = \nabla \cdot (|x|^\beta \nabla u) - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - \mu +…

The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = \Delta v - \mu + u, \qquad \int_\Omega v=0, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega…

We consider a parabolic-ODE-parabolic chemotaxis system with radially symmetric initial data in a two-dimensional disk under the $0$-Neumann boundary condition. Although our system shares similar mathematical structures as the Keller--Segel…

Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v =…

This paper is concerned with a quasilinear chemotaxis model with indirect signal production, $u_t = \nabla\cdot(D(u)\nabla u - S(u)\nabla v)$, $v_t = \Delta v - v + w$ and $w_t = \Delta w - w + u$, posed on a bounded smooth domain…

We study the Neumann initial-boundary value problem for the fully parabolic Keller-Segel system u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in\Omega, \ t>0, [1mm] v_t=\Delta v-v+u, \qquad x\in\Omega, \ t>0, where $\Omega$ is a ball in…

For the Keller-Segel system \[ \left\{\, \begin{aligned} u_t &= \Delta u - \nabla \cdot ( u \nabla v ), \\ v_t &= \Delta v - v + u \end{aligned} \right. \tag{$\star$} \] posed in a planar domain $\Omega$ with Neumann boundary conditions,…

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,…