相关论文: A double critical mass phenomenon in a no-flux-Dir…
Consider a class of chemotaxis-fluid model incorporating a volume-filling effect in the sense of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501-543), which is a supercritical parabolic-elliptic Keller-Segel system. As shown by…
We investigate the following two-species chemotaxis system with two chemicals involving flux-limitation \begin{align}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot \left(u(1+|\nabla v|^2)^{-\frac{p}{2}}\nabla v\right), & x \in…
The paper is concerned with the following chemotaxis system with nonlinear motility functions \begin{equation}\label{0-1}\tag{$\ast$} \begin{cases} u_t=\nabla \cdot (\gamma(v)\nabla u- u\chi(v)\nabla v)+\mu u(1-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…
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…
Unboundedness of solutions is shown to occur in a one-dimensional quasilinear parabolicparabolic chemotaxis system for any initial mass. Our result is also independent of the relation between the speeds of the diffusion of cells and…
This paper is concerned with the Keller--Segel system with flux limitation, \begin{align} \tag{$\ast$} \begin{cases} u_t=\Delta u - \nabla \cdot (uf(|\nabla v|^{2})\nabla v), \\ v_t=\Delta v - v + u \end{cases} \end{align} in bounded…
We consider nonnegative radially symmetric solutions of the parabolic-elliptic Keller-Segel system \begin{align*} \left\lbrace \begin{array}{r@{}l@{\quad}l} &u_t=\Delta u-\nabla \cdot \big(u\nabla v\big),\\ &0=\Delta v -\mu + u , \\…
We consider the Hamiltonian system with Neumann boundary conditions: \[ -\Delta u + \mu u=v^{q }, \quad -\Delta v+ \mu v=u^{p} \quad \text{ in $\Omega$}, \qquad u, v >0 \quad \text{ in $\Omega$,} \qquad \partial_\nu u= \partial_\nu v=0…
Perhaps the most classical diffusion model for chemotaxis is the Keller-Segel system $\begin{equation} \begin{cases} u_{t} =\Delta u - \nabla \cdot(u \nabla v) \ \ \ \text{in } \mathbb{R}^2\times(0,T),\\[5pt] v =…
This work deals with a fully parabolic chemotaxis model with nonlinear production and chemoattractant. The problem is formulated on a bounded domain and, depending on a specific interplay between the coefficients associated to such…
In this paper we consider radially symmetric solutions of the following parabolic--elliptic cross-diffusion system \begin{equation*} \begin{cases} u_t = \Delta u - \nabla \cdot (u f(|\nabla v|^2 )\nabla v) + g(u), & \\[2mm] 0= \Delta v…
In this study, we explore the quasilinear two-species chemotaxis system with two chemicals \begin{align}\tag{$\star$} \begin{cases} u_t = \nabla \cdot(D(u)\nabla u) - \nabla \cdot \left(S(u) \nabla v\right), & x \in \Omega, \ t > 0, \\ 0 =…
We carry on our studies related to the fully parabolic quasilinear Keller-Segel system started in [6] and continued in [7]. In the above mentioned papers we proved finite-time blowup of radially symmetric solutions to the quasilinear…
This paper is concerned with radially symmetric solutions of the parabolic-elliptic version of the Keller-Segel system with flux limitation, as given by \begin{equation} \left\{ \begin{array}{l} \displaystyle u_t=\nabla \cdot…
The evolution of a chemotactic system involving a population of cells attracted to self-produced chemicals is described by the Keller-Segel system. In spacial dimension 2, this system demonstrates a balance between the spreading effect of…
In this paper we consider the zero-flux chemotaxis-system \begin{equation*} \begin{cases} u_{t}=\Delta u-\nabla \cdot (u \chi(v)\nabla v) & \textrm{in}\quad \Omega\times (0,\infty), \\ 0=\Delta v-v+g(u) & \textrm{in}\quad \Omega\times…
We consider a model of cell motion with boundary signal production which describes some aspects of eukaryotic cell migration. Generic polarity markers located in the cell are transported by actin which they help to polymerize. This leads to…
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}…
In this paper we consider a one-dimensional fully parabolic quasilinear Keller-Segel system with critical nonlinear diffusion. We show uniform-in-time boundedness of solutions, which means, that unlike in higher dimensions, there is no…