Related papers: Chemotaxis can prevent thresholds on population de…

We consider a two-species chemotaxis model in $\R^d(d \ge 3)$ featuring nonlinear porous medium-type diffusion and nonlocal attractive power-law interaction. Here, the nonlinear diffusion is chosen to be $1/m_1+1/m_2=(d+2)/d$ in such a way…

In this paper we study the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{ t}=\nabla \cdot ((u+1)^{m-1} \nabla u-(u+1)^\alpha \chi(v)\nabla v) + ku-\mu u^2 & x\in \Omega, t>0, \\ v_{t} = \Delta v-vu & x\in \Omega, t>0,\\…

For given total mass $m>0$ we show unique solvability of the stationary chemotaxis-consumption model \[ \begin{cases} 0= \Delta u - \chi \nabla \cdot (\frac{u}{v} \nabla v) \\ 0= \Delta v - uv \\ \int_\Omega u = m \end{cases} \] under…

This article deals with an initial-boundary value problem for the coupled chemotaxis-haptotaxis system with nonlinear diffusion \begin{align*} u_t=&\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu…

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…

This paper is concerned with the following parabolic-parabolic-elliptic chemotaxis system with singular sensitivity and Lotka-Volterra competitive kinetics, \begin{equation} \begin{cases} u_t=\Delta u-\chi_1 \nabla\cdot (\frac{u}{w} \nabla…

We consider the following chemotaxis systems $$\begin{cases}u_t=\Delta u-\chi_1\nabla(u\nabla v_1)+\chi_2\nabla(u\nabla v_2)+u(a-bu),\ \ x\in\mathbb R^N,t>0,\\0=(\Delta-\lambda_1I)v_1+\mu_1u,\ \ x\in\mathbb…

Assuming that $0<\chi<\sqrt{\frac{2}n}$, $\kappa\ge 0$ and $\mu>\frac{n-2}{n}$, we prove global existence of classical solutions to a chemotaxis system slightly generalizing \[ \begin{split} u_t &= \Delta u - \chi \nabla\cdot ( \frac{u}{v}…

Introducing a suitable solution concept, we show that in bounded smooth domains $\Omega\subset \mathbb{R}^n$, $n\ge 1$, the initial boundary value problem for the chemotaxis system \begin{align*} u_t&=\Delta u…

We consider the chemotaxis model \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v), \\ v_t = \Delta v - vw, \\ w_t = -\delta w + u \end{cases} \end{align*} in smooth, bounded domains $\Omega \subset \mathbb R^n$, $n…

In present paper, we consider a chemotaxis consumption system with density-signal governed sensitivity and logistic source: $u_t=\Delta u-\nabla\cdot(\frac{S(u)}{v}\nabla v)+ru-\mu u^2$, $v_t=\Delta v-uv$ in a smooth bounded domain…

In the present review we focus on the chemotaxis-consumption model $\partial_t u - \Delta u = - \nabla \cdot (u \nabla v)$ and $\partial_t v - \Delta v = - u^s v$ in $(0,T) \times \Omega$, for any fixed $s \geq 1$, endowed with isolated…

This work studies the existence of classical solutions to a class of chemotaxis systems reading \[\begin{cases} u_t = \Delta u-\chi \nabla\cdot(u \nabla v) + \mu_1 u^k -\mu_2 u^{k+1}, & \text{in} \; \Omega\times(0,T_{\text{max}}), \\ v_t=…

We study this zero-flux attraction-repulsion chemotaxis model, with linear and superlinear production $g$ for the chemorepellent and sublinear rate $f$ for the chemoattractant: \begin{equation}\label{problem_abstract} \tag{$\Diamond$}…

This paper deals with the problem of global solvability and boundedness of classical solutions to a fully parabolic chemotaxis system with singular sensitivity in any dimensional setting. In particular, We show that the system…

This paper is concerned with the Neumann initial-boundary value problem for the two-species chemotaxis system with consumption of chemoattractant \begin{equation*} u_t=\Delta u-\chi_1\nabla\cdot(u\nabla w), \end{equation*} \begin{equation*}…

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 consider the system \begin{align*} \begin{cases} n_t + u \cdot \nabla n = \Delta n - \chi \nabla \cdot (\frac{n}{c} \nabla c),

This paper deals with unbounded solutions to a class of chemotaxis systems. In particular, for a rather general attraction-repulsion model, with nonlinear productions, diffusion, sensitivities and logistic term, we detect Lebesgue spaces…

In this work we investigate the following chemo-attraction with consumption model in bounded domains of \, $\mathbb{R}^N$ ($N=1,2,3$): $$ \partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v $$…