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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 solution of following chemotaxis system with competitive kinetics and nonlocal terms \begin{eqnarray*} \left\{ \begin{array}{llll} u_t=d_1\Delta u-\chi_1\nabla\cdot(u\nabla w)+u\left(a_0-a_1u-a_{2}v-a_3\int_\Omega…

\indent In this paper, we study a class of parabolic-elliptic Keller-Segel systems with diffusion sensitivity dependent on spatial position, given by type \begin{equation} \left\{ \begin{array}{ll} u_{t} = \bigtriangledown\cdot(|x|^{\beta}…

This paper deals with a boundary-value problem for a coupled chemotaxis-Navier-Stokes system involving tensor-valued sensitivity with saturation $$\left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad…

In this work, we study the doubly degenerate nutrient taxis system with logistic source \begin{align} \begin{cases}\tag{$\star$}\label{eq 0.1} u_t=\nabla \cdot(u^{l-1} v \nabla u)- \nabla \cdot\left(u^{l} v \nabla v\right)+ u - u^2, \\…

We consider a degenerate quasilinear chemotaxis--Stokes type involving rotation in the aggregative term, \begin{equation} \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n^m-\nabla\cdot(nS(x,n,c)\cdot\nabla c),\quad x\in \Omega, t>0,…

The parabolic-elliptic cross-diffusion system \[ \left\{ \begin{array}{l} u_t = \Delta u - \nabla \cdot \Big(uf(|\nabla v|^2) \nabla v \Big), \\[1mm] 0 = \Delta v - \mu + u, \qquad \int_\Omega v=0, \qquad \mu:=\frac{1}{|\Omega|} \int_\Omega…

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.…

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 deals with the classical solution of the following chemotaxis system with generalized logistic growth and indirect signal production \begin{eqnarray} \left\{ \begin{array}{llll} & u_t=\epsilon\Delta u-\nabla\cdot(u\nabla…

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=…

We study a chemotaxis-Stokes system with signal consumption and logistic source terms of the form \noindent \begin{align*} \left\{ \begin{array}{r@{\ }l@{\quad}l@{\quad}l@{\,}c} n_{t}+u\cdot\!\nabla n&=\Delta n-\nabla\!\cdot(n\nabla…

We study the Neumann initial-boundary problem for the chemotaxis system \begin{align*} \left\{\begin{array}{c@{\,}l@{\quad}l@{\,}c} u_{t}&=\Delta u-\nabla\!\cdot(u\nabla v),\ &x\in\Omega,& t>0,\\ v_{t}&=\Delta v-v+u+f(x,t),\ &x\in\Omega,&…

In this paper, we introduce the nonlinear diffusion term $\nabla\cdot(D(u)\nabla u)$ into the chemotaxis-May-Nowak model to investigate the effects of $D(u)$ and chemotaxis on the global existence, boundedness, and finite time blow-up of…

Derived from a biophysical model for the motion of a crawling cell, the system \[(*)~\begin{cases}u_t=\Delta u-\nabla\cdot(u\nabla v)\\0=\Delta v-kv+u\end{cases}\] is investigated in a finite domain $\Omega\subset\mathbb{R}^n$, $n\geq2$,…

We study the chemotaxis model $\partial$ t u = div($\nabla$u -- u$\nabla$w) + $\theta$v -- u in (0, $\infty$) x $\Omega$, $\partial$ t v = u -- $\theta$v in (0, $\infty$) x $\Omega$, $\partial$ t w = D$\Delta$w -- $\alpha$w + v in (0,…

We consider a parabolic-elliptic system of partial differential equations with chemotaxis and logistic growth given by the system $$ \left\{ \begin{array}{l} u_t -\Delta (u \gamma(v)= \mu u(1-u), \\ - \Delta v +v=u, \end{array} \right. $$…

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…

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,…

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…