相关论文: Boundedness in a chemotaxis system with consumed c…

This work considers the Keller-Segel consumption system \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=\Delta (u\phi(v))+au-bu^\gamma,\quad &x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad &x\in\Omega,\quad t>0 \end{array} \right.…

The current series of three papers is concerned with the asymptotic dynamics in the following chemotaxis model $$\partial_tu=\Delta u-\chi\nabla(u\nabla v)+u(a(x,t)-ub(x,t))\ ,\ 0=\Delta v-\lambda v+\mu u \ \ (1)$$where $\chi, \lambda, \mu$…

This paper investigates the spreading properties of globally defined bounded positive solutions of a chemotaxis system featuring a logistic source and consumption: \[ \left\{ \begin{aligned} &\partial_tu=\Delta u - \chi\nabla\cdot(u\nabla…

In a smoothly bounded convex domain $\Omega\subset R^n$ with $n\ge 1$, a no-flux initial-boundary value problem for \[ \left\{ \begin{array}{l} u_t=\Delta \big(u\phi(v)\big), v_t=\Delta v-uv, \end{array} \right. \] is considered under the…

In this paper, we study a nutrient-taxis model with porous medium slow diffusion \begin{align*} \left\{ \begin{aligned} &u_t=\Delta u^m-\chi\nabla\cdot(u\nabla v)+\xi uv-\rho u, \\ &v_t-\Delta v=-vu+\mu v(1-v), \end{aligned}\right.…

We study a chemotaxis-consumption mechanism, in which some chemical signal and cells density interact each other. In order to control the concentration of such a population, sources involving gradient nonlinearities, which introduce a…

We are concerned with the following full Attraction-Repulsion Keller-Segel (ARKS) system \begin{equation}\label{ARKS}\tag{$\ast$} \begin{cases} u_t=\Delta u-\nabla\cdot(\chi u\nabla v)+\nabla\cdot(\xi u\nabla w), &x\in \Omega, ~~t>0,…

We study positive solutions to the steady state reaction diffusion systems of the form: \begin{equation} \left\{\begin{array}{ll} -\Delta u = \lambda f(v)+\mu h(u), & \Omega,\\ -\Delta v = \lambda g(u)+\mu q(v),& \Omega,\\ \frac{\partial…

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…

In this paper, we study traveling wave solutions of the chemotaxis systems \begin{equation} \begin{cases} u_{t}=\Delta u -\chi_1\nabla( u\nabla v_1)+\chi_2 \nabla(u\nabla v_2 )+ u(a -b u), \qquad \ x\in\mathbb{R} \\…

In this paper, we consider the following system $$\left\{\begin{array}{ll} n_t+u\cdot\nabla n&=\Delta n-\nabla\cdot(n\mathcal{S}(|\nabla c|^2)\nabla c)-nm,\\ c_t+u\cdot\nabla c&=\Delta c-c+m,\\ m_t+u\cdot\nabla m&=\Delta m-mn,\\ u_t&=\Delta…

In this article, we study domains $\Omega \subset \mathbb{S}^2$ that support positive solutions of the overdetermined problem $$ \Delta u + f(u,|\nabla u|)=0 \quad \text{in } \Omega, $$ subject to the boundary conditions $u=0$ on…

This work studies the following system of parabolic partial differential equations \begin{equation*} \begin{cases} \displaystyle \frac{\partial u}{\partial t} = D\Delta u + \chi \nabla \cdot(u \nabla v) + ru(1-u) - u v, \quad & x \in…

This paper deals with a lower bound for the blow-up time for solutions of the fully parabolic chemotaxis system \begin{equation*} \begin{cases} u_t=\nabla \cdot [(u+\alpha)^{m_1-1} \nabla u-\chi u(u+\alpha)^{m_2-2} \nabla v] & {\rm in} \;…

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 +…

We consider a chemotaxis-fluid system involving nonlinear cell diffusion of porous medium type, signal consumption by cells, and rather general, possibly matrix-valued, chemotactic sensitivities. It is shown that if the corresponding…

We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $…

We prove existence of global weak solutions to the chemotaxis system $ u_t=\Delta u - \nabla\cdot (u\nabla v) +\kappa u -\mu u^2 $ $ v_t=\Delta v-v+u $ under homogeneous Neumann boundary conditions in a smooth bounded convex domain…

We consider the heat equation in a smooth bounded convex domain $\Omega \subset \mathbb{R}^2$ with nonlinear Neumann boundary condition $\partial_\nu u = \lambda (u - u^3)$. Stable non-constant stationary solutions do not exist when…

We investigate the uniform regularity and vanishing viscosity limit for the incompressible chemotaxis-Navier-Stokes system in a smooth bounded domain $\Omega\subset\mathbb{R}^3$. It is shown that there exists a unique strong solution of the…