相关论文: Boundedness in a Keller-Segel system with external…

We study the Neumann initial-boundary problem for the chemotaxis system $$ \left\{\begin{array}{ll} u_t= \Delta u - \nabla \cdot (u\nabla v), & x\in \Omega, \, t>0, 0=\Delta v - \mu(t)+w, & x\in \Omega, \, t>0, \tau w_t + \delta w = u, &…

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 following chemotaxis system under homogeneous Neumann boundary conditions in a smooth, open, bounded domain $\Omega \subset \mathbb{R}^n$ with $n \geq 3$: \begin{equation*} \begin{cases} u_t = \Delta u - \chi \nabla \cdot…

This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such…

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.…

This work studies the chemotaxis-haptotaxis system $$\left\{ \begin{array}{ll} u_t= \Delta u - \chi \nabla \cdot (u\nabla v) - \xi \nabla \cdot (u\nabla w) + \mu u(1-u-w), &\qquad x\in \Omega, \, t>0, \\[1mm] v_t=\Delta v-v+u, &\qquad x\in…

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 -…

This paper studies the chemotaxis-haptotaxis system \begin{equation}\nonumber \left\{ \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w)+\mu u(1-u-w), &(x,t)\in \Omega\times (0,T),\\ v_t=\Delta v-v+u,…

This paper deals with the Neumann boundary value problem for the system $$u_t=\nabla\cdot\left(D(u)\nabla u\right)-\nabla\cdot\left(S(u)\nabla v\right)+f(u) ,\quad x\in\Omega,\ t>0$$ $$v_t=\Delta v-v+u,\quad x\in\Omega,\ t>0$$ in a smooth…

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 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 boundary value problem $-\Delta u + u =\lambda e^u$ in $\Omega$ with Neumann boundary condition, where $\Omega$ is a bounded smooth domain in $\mathbb R^2$, $\lambda>0.$ This problem is equivalent to the stationary…

This paper deals with classical solutions to the parabolic-parabolic system \begin{align*} \begin{cases} u_t=\Delta (\gamma (v) u ) &\mathrm{in}\ \Omega\times(0,\infty), \\[1mm] v_t=\Delta v - v + u &\mathrm{in}\ \Omega\times(0,\infty),…

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…

We consider the chemotaxis system \begin{eqnarray*} \begin{cases} \begin{array}{lll} \medskip u_t =\Delta u^m - \nabla(\frac{u}{v}\nabla v),&{} x\in\Omega,\ t>0, \medskip v_t =\Delta v -uv,&{}x\in\Omega,\ t>0, \medskip \frac{\partial…

This paper deals with the homogeneous Neumann boundary-value problem for the chemotaxis-consumption system \begin{eqnarray*} \begin{array}{llc} u_t=\Delta u-\chi\nabla\cdot (u\nabla v)+\kappa u-\mu u^2,\\ v_t=\Delta v-uv, \end{array}…

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…

The chemotaxis system \begin{align*} u_t &= \Delta u - \nabla \cdot (u\nabla v), \\ v_t &= \Delta v - uv, \end{align*} is considered under the boundary conditions $\frac{\partial u}{\partial\nu}- u\frac{\partial v}{\partial\nu}=0$ and…

We study the following Neumann boundary problem related to the stationary solutions of the Keller-Segel system, a basic model of chemotaxis phenomena: \[ \left\{\begin{array}{ll} -\Delta_g u +\beta u =\lambda\left(\frac{Ve^u}{\int_{\Sigma}…

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 +…