Related papers: Boundedness in a two-dimensional chemotaxis-haptot…

Nonnegative solutions of the Neumann initial-boundary value problem for the chemotaxis system \begin{align}\label{prob:star}\tag{$\star$} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) + \lambda u - \mu u^\kappa, \\\\ 0 = \Delta v…

We consider an initial-boundary value problem for the incompressible chemotaxis-Navier-Stokes equations generalizing the porous-medium-type diffusion model $ \quad n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(n\chi(c)\nabla c), $ $ \quad…

In this paper we consider nonnegative solutions of the following parabolic-elliptic cross-diffusion system \begin{equation*} \left\{ \begin{array}{l} \begin{aligned} &u_t = \Delta u - \nabla(u f(|\nabla v|^2 )\nabla v), \\[6pt] &0= \Delta v…

We study the chemotaxis-fluid system \begin{align*} \left\{\begin{array}{r@{\,}l@{\quad}l@{\,}c} n_{t}&=\Delta n-\nabla\!\cdot(n\nabla c)-u\cdot\!\nabla n,\ &x\in\Omega,& t>0,\\ c_{t}&=\Delta c-c+f(n)-u\cdot\!\nabla c,\ &x\in\Omega,& t>0,\\…

We consider the chemotaxis-consumption system \[ \left\{ \begin{aligned} u_t &= \Delta u - \chi \nabla \cdot (u\nabla v) \\ v_t &= \Delta v - uv \end{aligned} \right. \] in a smooth bounded domain $\Omega \subseteq \mathbb{R}^n$, $n \geq…

We introduce a novel gradient-based damping term into a Keller-Segel type taxis model with motivation from ecology and consider the following system equipped with homogeneous Neumann-boundary conditions: \begin{equation} \begin{cases} u_t=…

This paper studies the dynamical behavior of classical solutions to a hyperbolic system of balance laws, derived from a chemotaxis model with logarithmic sensitivity, subject to time-dependent boundary conditions. It is shown that under…

In bounded $n$-dimensonal domains with $n\ge 1$, this manuscript examines an initial-boundary value problem for the system \[ \left\{ \begin{array}{l} u_{tt} = \nabla \cdot (\gamma(\Theta) \nabla u_t) + a \nabla \cdot (\gamma(\Theta) \nabla…

In this paper, we study the initial boundary value problem of the important hyperbolic Kirchhoff equation $$u_{tt}-\left(a \int_\Omega |\nabla u|^2 \dif x +b\right)\Delta u = \lambda u+ |u|^{p-1}u ,$$ where $a$, $b>0$, $p>1$, $\lambda \in…

In this paper we focus on an attraction-repulsion chemotaxis model with consumed signals, formulated in bounded and smooth domains $\Omega$ of $\R^n$, with $n\geq 2$. We derive sufficient conditions on its data yielding global and bounded…

In this paper, we investigate a chemotaxis-fluid system involving both the effect of potential force on cells and the effect of chemotactic force on fluid: \begin{equation*} \left\{ \begin{split} \partial_t n + \mathbf{u}\cdot\nabla n & =…

In this paper we study the initial boundary value problem for the system $\Delta v= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla…

The present work proceeds to consider the convergence of the solutions to the following doubly degenerate chemotaxis-consumption system \begin{align*} \left\{ \begin{array}{r@{\,}l@{\quad}l@{\,}c} &u_{t}=\nabla\cdot\big(u^{m-1}v\nabla…

This paper deals with the homogeneous Neumann boundary-value problem for the Keller--Segel system \begin{align*} \begin{cases} u_t=\Delta u - \chi \nabla \cdot (u|\nabla v|^{p-2}\nabla v),\\[] v_t=\Delta v - v + u^{\theta} \end{cases}…

A three-dimensional chemotaxis-Navier-Stokes system is considered. It is known that for all suitably regular initial data, a corresponding initial-boundary value problem admits at least one global weak solution which can be obtained as the…

In this paper, we study the consumption-chemotaxis-Stokes model with porous medium slow diffusion in a three dimensional bounded domain with zero-flux boundary conditions and no-slip boundary condition. In recent ten years, many efforts…

We investigate the Keller--Segel--(Navier--)Stokes system posed in a smooth bounded domain \(\Omega \subset \mathbb{R}^N\) with \(N = 2,3\): \begin{equation*} \begin{cases} n_t + u \cdot \nabla n = \Delta n - \nabla \cdot \big( n S(n)\nabla…

This paper studies a fractional attraction-repulsion system with generalized logistic source and nonlinear productions: \begin{equation*} \left\{ \begin{aligned} &u_t = -(-\Delta)^\alpha u - \chi_1 \nabla \cdot (u \nabla v) + \chi_2 \nabla…

This paper deals with a problem which describes tuberculosis granuloma formation \begin{align*} \begin{cases} u_t = \Delta u - \nabla \cdot (u \nabla v) - uv - u + \beta, &x \in \Omega,\ t>0, \\ v_t = \Delta v + v -uv + \mu w, &x \in…

It has been well established that, in attraction-repulsion Keller-Segel systems of the form\begin{equation*} \left\{ \begin{aligned} u_t &= \Delta u - \chi \nabla \cdot (u\nabla v) + \xi \nabla \cdot (u\nabla w), \\ \tau v_t &= \Delta v +…