Related papers: Boundedness in a chemotaxis system with consumed c…

We show global existence and boundedness of classical solutions to a virus infection model with chemotaxis in bounded smooth domains of arbitrary dimension and for any sufficiently regular nonnegative initial data and homogeneous Neumann…

In this paper, we consider the following parabolic-parabolic-elliptic system } \begin{align*} \left\{\aligned & u_t=\Delta u-\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+au-\mu u^{\alpha}, && x\in\Omega, t>0,\\ & v_t=\Delta…

This paper is concerned with the attraction-repulsion chemotaxis system with superlinear logistic degradation, \begin{align*} \begin{cases} u_t = \Delta u - \chi \nabla\cdot(u \nabla v) + \xi \nabla\cdot (u \nabla w) + \lambda u - \mu u^k,…

In a ball $\Omega\subset R^n$ with arbitrary $n\ge 1$, the chemotaxis-consumption system \[ \left\{ \begin{array}{l} u_t = \nabla \cdot \big(D(u)\nabla u\big) - \nabla \cdot (u\nabla v), \\[1mm] 0 = \Delta v - uv, \end{array} \right. \] is…

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…

In this work we investigate the following chemo-attraction with consumption model in bounded domains of \, $\mathbb{R}^N$ ($N=1,2,3$): $$ \partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v $$…

This paper deals with the fully parabolic chemotaxis-convection model with sensitivity functions for tumor angiogenesis, \begin{align*} \begin{cases} u_t=\Delta u-\nabla \cdot (u\chi_1(v)\nabla v) +\nabla \cdot (u\chi_2(w)\nabla w), &x \in…

In this study, we consider the following extended attraction chemotaxis system of two species parabolic-parabolic-elliptic type with nonlocal terms \[ \begin{cases} u_t=d_1\Delta u-\chi_1\nabla (u\cdot \nabla…

We revisit the following chemotaxis system modeling tumor invasion \begin{equation*} \begin{cases} u_t=\Delta u-\nabla \cdot(u\nabla v),& x\in\Omega, t>0,\\ v_t=\Delta v+wz,& x\in\Omega, t>0,\\ w_t=-wz,& x\in\Omega, t>0,\\ z_t=\Delta z-z+u,…

We show the existence of locally bounded global solutions to the chemotaxis system \[ u_t = \nabla\cdot(D(u)\nabla u) - \nabla\cdot(\frac{u}{v} \nabla v) \] \[ v_t = \Delta v - uv \] with homogeneous Neumann boundary conditions and suitably…

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…

In the present work we investigate an optimal control problem related to the following chemotaxis-consumption model in a bounded domain $\Omega\subset \mathbb{R}^3$: $$\partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t…

In this paper, the indirect signal production system with nonlinear transmission is considered \[ \left\{ \begin{array}{lll} & u_t = \Delta u-\nabla\cdot(u \nabla v), \\ \displaystyle & v_t =\Delta v-v+w,\\ \displaystyle & w_t =\Delta w-w+…

In a smoothly bounded domain $\Omega \subset \mathbb{R}^N$ $(N\in \mathbb{N})$, a no-flux initial-boundary value problem for the degenerate chemotaxis system with volume-filling effects, \begin{align*} u_t = \nabla \cdot (D(u,v) \nabla u -…

The chemotaxis--Navier--Stokes system \begin{equation*}\label{0.1} \left\{\begin{array}{ll} n_t+u\cdot \nabla n=\triangle n-\chi\nabla\cdotp \left(\displaystyle\frac n {c}\nabla c\right)+n(r-\mu n), c_t+u\cdot \nabla c=\triangle c-nc, u_t+…

This paper investigates the following chemotaxis system featuring weak degradation and nonlinear motility functions \begin{equation}\label{Model1} \begin{cases} u_{t} = (\gamma(v)u)_{xx} + r - \mu u, & x \in [0,L],\ t > 0, v_{t} = v_{xx} -…

This paper studies the asymptotic behavior of solutions of the parabolic-parabolic chemotaxis model with logistic-type sources in heterogeneous bounded domains: \begin{equation*} \label{u-v-eq00} \begin{cases} u_t=\Delta u-\chi\nabla\cdot…

This paper deals with a boundary-value problem for a coupled quasilinear chemotaxis--haptotaxis model with nonlinear diffusion $$\left\{\begin{array}{ll} u_t=\nabla\cdot(D(u)\nabla u)-\chi\nabla\cdot(u\nabla v)-\xi \nabla\cdot(u\nabla…

This paper is Part II of a series on global existence and asymptotic behavior of positive solutions to \begin{equation*} \begin{cases} \displaystyle u_t=\Delta u-\chi_0\nabla\cdot\left(\frac{u^m}{(1+v)^\beta}\nabla…

We show that the attraction-repulsion chemotaxis system \begin{equation*} \begin{cases} u_t = \Delta u - \chi\nabla\cdot(u\nabla v_1) + \xi\nabla\cdot(u\nabla v_2)\\ \partial_t v_1 = \Delta v_1 - \beta v_1 + \alpha u \\ \partial_t v_2 =…