Boundedness in a fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity

This paper deals with the quasilinear fully parabolic attraction–repulsion chemotaxis system $\left\{\begin{array}{cc}{u}_t=\nabla \cdot (D\left(u\right)\nabla u)-\nabla \cdot (G\left(u\right)\chi \left(v\right)\nabla v)+\nabla \cdot (H\left(u\right)\xi \left(w\right)\nabla w),\hfill & x\in \Omega ,t>0,\hfill \\ v_t=d_1\Delta v+\alpha u-\beta v,\hfill & x\in \Omega ,t>0,\hfill \\ w_t=d_2\Delta w+\gamma u-\delta w,\hfill & x\in \Omega ,t>0,\hfill \end{array}\right.$under homogeneous Neumann boundary conditions and initial conditions, where $\Omega \subset {\mathbb{R}}^n$ $(n\ge 1)$ is a bounded domain with smooth boundary, $d_1,d_2,\alpha ,\beta ,\gamma ,\delta >0$ are constants. Also, $D,G,H\in C^2\left([0,\infty )\right)$ fulfill that $a_0{(s+1)}^{m-1}\le D\left(s\right)\le a_1{(s+1)}^{m-1}$ with $a_0,a_1>0$ and $m\in \mathbb{R}$; $G\left(0\right)=0$, $0\le G\left(s\right)\le b_0{(s+1)}^{q-1}$ with $b_0>0$ and $q<min\{2,m+1\}$; $H\left(0\right)=0$, $0\le H\left(s\right)\le c_0{(s+1)}^{r-1}$ with $c_0>0$ and $r<min\{2,m+1\}$, and $\chi ,\xi$ satisfy that $0\le \chi \left(s\right)\le \frac{{\chi }_0}{s^{k_1}}$ with ${\chi }_0>0$ and $k_1>1$; $0\le \xi \left(s\right)\le \frac{{\xi }_0}{s^{k_2}}$ with ${\xi }_0>0$ and $k_2>1$. Global existence and boundedness in the case that $w=0$ were proved by Ding (2018). However, there is no work on the above fully parabolic attraction–repulsion chemotaxis system with nonlinear diffusion and signal-dependent sensitivity. This paper develops global existence and boundedness of classical solutions to the above system.     

Boundedness in a higher‐dimensional chemotaxis system with porous medium diffusion and general sensitivity

This paper deals with the following chemotaxis system: in a bounded domain with smooth boundary under no‐flux boundary conditions, where satisfies for all with l ?2 and some nondecreasing function on [0,∞ ). Here, f (v )∈C ¹([0,∞ )) is nonnegative for all v ?0. It is proved that when , the system possesses at least one global bounded weak solution for any sufficiently smooth nonnegative initial data. This extends a recent result by Wang (Math. Methods Appl. Sci. 2016 39 : 1159–1175) which shows global existence and boundedness of weak solutions under the condition . Copyright © 2017 John Wiley & Sons, Ltd.     

Boundedness in a parabolic‐elliptic chemotaxis system with nonlinear diffusion and sensitivity and logistic source

In this paper, we study the zero‐flux chemotaxis‐system where Ω is a bounded and smooth domain of , n≥1, and where , k,μ>0 and α≤1. For any v≥0, the chemotactic sensitivity function is assumed to behave as the prototype χ(v)=χ₀/(1+av)², with a≥0 and χ₀>0. We prove that for any nonnegative and sufficiently regular initial data u(x,0), the corresponding initial‐boundary value problem admits a unique global bounded classical solution if α<1; indeed, for α=1, the same conclusion is obtained provided μ is large enough. Finally, we illustrate the range of dynamics present within the chemotaxis system in 1, 2, and 3 dimensions by means of numerical simulations.     

Boundedness in a fully parabolic chemotaxis‐consumption system with nonlinear diffusion and sensitivity,and logistic source

In this paper we study the zero‐flux chemotaxis‐system Ω being a convex smooth and bounded domain of , , and where , and . For any the chemotactic sensitivity function is assumed to behave as the prototype , with and . We prove that for nonnegative and sufficiently regular initial data and , the corresponding initial‐boundary value problem admits a unique globally bounded classical solution provided μ is large enough.     

Boundedness and large time behavior for a chemotaxis system with signal-dependent motility and indirect signal consumption

In this paper, we study the following chemotaxis system with signal-dependent motility, indirect signal consumption and logistic source $\left\{\begin{array}{cc}{u}_t=\Delta \left(u\gamma \left(v\right)\right)+\rho u-\mu u^l,\hfill & x\in \Omega ,t>0,\hfill \\ v_t=\Delta v-vw,\hfill & x\in \Omega ,t>0,\hfill \\ w_t=-\delta w+u,\hfill & x\in \Omega ,t>0\hfill \end{array}\right.$under homogeneous Neumann boundary conditions in a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$, where the motility function $\gamma \left(v\right)\in C^3\left((0,+\infty )\right),\gamma \left(v\right)>0,{\gamma }^{\prime }\left(v\right)<0$ on $(0,+\infty )$, ${lim}_{v\to \infty }\gamma \left(v\right)=0$, $\rho >0,\mu >0,l>1$ and $\delta >0$. The purpose of this paper is to prove that if $l>max\{1,\frac{n}{2}\}$, then the system possesses a global solution. In addition, if $l$ satisfies $l\left\{\begin{array}{cc}\ge 2,\hfill & \text{if }n\le 3,\hfill \\ >\frac{n}{2},\hfill & \text{if }n\ge 4,\hfill \end{array}\right.$then the solution $(u,v,w)$ satisfies ${6u(\cdot ,t)-{\left(\frac{\rho }{\mu }\right)}^{\frac{1}{l-1}}6}_{L^{\infty }\left(\Omega \right)}+{6v(\cdot ,t)6}_{L^{\infty }\left(\Omega \right)}+{6w(\cdot ,t)-\frac{1}{\delta }{\left(\frac{\rho }{\mu }\right)}^{\frac{1}{l-1}}6}_{L^{\infty }\left(\Omega \right)}$$\to 0ast\to \infty .$     

Global existence and boundedness in a chemotaxis-Stokes system with arbitrary porous medium diffusion

In this short paper, we establish the global existence and boundedness of solutions to the initial-boundary value problem of a chemotaxis-Stokes system with porous-medium-like cell diffusion Δn^m for all adiabatic exponents m>1. Our result extend the corresponding result under the constraint $m\in (\frac{3}{2},2]$.     

Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement

A class of chemotaxis-Stokes systems generalizing the prototype

$\{\begin{array}{ccc}\hfill n_t+u?\mathrm{?}n& \hfill =\hfill & \mathrm{?}?\left(n^{m?1}\mathrm{?}n\right)?\mathrm{?}?\left(n\mathrm{?}c\right),\hfill \\ \hfill c_t+u?\mathrm{?}c& \hfill =\hfill & \mathrm{\Delta }c?nc,\hfill \\ \hfill u_t+\mathrm{?}P& \hfill =\hfill & \mathrm{\Delta }u+n\mathrm{?}?,\mathrm{?}?u=0,\hfill \end{array}$

is considered in bounded convex three-dimensional domains, where $?\in W^{2,\infty }(\mathrm{\Omega })$ is given.The paper develops an analytical approach which consists in a combination of energy-based arguments and maximal Sobolev regularity theory, and which allows for the construction of global bounded weak solutions to an associated initial-boundary value problem under the assumption that

(0.1)$m>\frac{9}{8}.$

Moreover, the obtained solutions are shown to approach the spatially homogeneous steady state $(\frac{1}{|\mathrm{\Omega }|}{\int }_{\mathrm{\Omega }}{n}_0,0,0)$ in the large time limit.This extends previous results which either relied on different and apparently less significant energy-type structures, or on completely alternative approaches, and thereby exclusively achieved comparable results under hypotheses stronger than (0.1).     

Boundedness in a two‐dimensional attraction–repulsion system with nonlinear diffusion

This paper is devoted to the attraction–repulsion chemotaxis system with nonlinear diffusion: where χ > 0, ζ > 0, α_i>0, β_i>0 (i = 1,2) and f(s)≤κ ? μs^τ. In two‐space dimension, we prove the global existence and uniform boundedness of the classical solution to this model for any μ > 0. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundedness in a chemotaxis system with nonlinear signal production

This work deals with the zero-Neumann boundary problem to a fully parabolic chemotaxis system with a nonlinear signal production function f(s) fulfilling 0 ≤ f(s) ≤ Ks~α for all s ≥ 0, where K and α are positive parameters. It is shown that whenever 0 α 2/n(where n denotes the spatial dimension) and under suitable assumptions on the initial data,this problem admits a unique global classical solution that is uniformly-in-time bounded in any spatial dimension. The proof is based on some a priori estimate techniques.     

Population dynamical behavior of a single-species nonlinear diffusion system with random perturbation

We consider a single-species stochastic logistic model with the population’s nonlinear diffusion between two patches. We prove the system is stochastically permanent and persistent in mean, and then we obtain sufficient conditions for stationary distribution and extinction. Finally, we illustrate our conclusions through numerical simulation.     

Boundedness in a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source

We study a quasilinear parabolic-parabolic chemotaxis system with nonlinear logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. By establishing proper a priori estimates we prove that, with both the diffusion function and the chemotaxis sensitivity function being positive, the corresponding initial boundary value problem admits a unique global classical solution which is uniformly bounded. The result of this paper is a generalization of that of Cao (2014).     

Asymptotic behavior of solutions of a nonlinear diffusion equation with a source term of general form

Chernogolovka, Moscow District. Translated from Sibirskii Matematicheskii Zhurnal, Vol. 30, No. 1, pp. 35–47, January–February, 1989.