Boundedness in a two‐species chemotaxis system

This paper is concerned with the following coupled chemotaxis system with homogeneous Neumann boundary conditions in a bounded domain Ω?Rⁿ(n≥2) with smooth boundary, where λ, χ₁, χ₂, μ₁, μ₂, a₁, a₂, b₁, and b₂ are supposed to be positive and τ = 0,1. In the case τ = 0, based on some energy estimates for both u and v, it is shown that for any parameters, the system possesses a unique globally bounded solution if n = 2. Moreover, when τ = 1, relying on a comparison principle, for a range of parameters, the existence of a unique global bounded classical solution of problem is established for any n≥2 if Ω is convex. Copyright © 2015 John Wiley & Sons, Ltd.     

Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect

We consider the elliptic-parabolic PDE system

     

A unified method for boundedness in fully parabolic chemotaxis systems with signal-dependent sensitivity

This paper deals with the Keller–Segel system where Ω is a bounded domain in with smooth boundary , ; χ is a nonnegative function satisfying for some and . In the case that and , Fujie 2 established global existence of bounded solutions under the condition . On the other hand, when , Winkler 14 asserted global existence of bounded solutions for arbitrary . However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary . Moreover, the condition for K when cannot connect with the condition when . The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for χ and to build a mathematical bridge between the cases and .     

Boundedness and asymptotic stability of solutions to a two-species chemotaxis system with consumption of chemoattractant

This paper deals with a two-competing-species chemotaxis system with consumption of chemoattractant

$\{\begin{array}{cc}{u}_t=d_1\mathrm{\Delta }u?\mathrm{?}?(u{\chi }_1(w)\mathrm{?}w)+{\mu }_{1}u(1?u?a_{1}v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ v_t=d_2\mathrm{\Delta }v?\mathrm{?}?(v{\chi }_2(w)\mathrm{?}w)+{\mu }_{2}v(1?a_{2}u?v),\hfill & x\in \mathrm{\Omega },t>0,\hfill \\ w_t=d_3\mathrm{\Delta }w?(\alpha u+\beta v)w,\hfill & x\in \mathrm{\Omega },t>0\hfill \end{array}$

under homogeneous Neumann boundary conditions in a bounded domain $\mathrm{\Omega }?{\mathbb{R}}^n$ ($n\ge 1$) with smooth boundary, where the initial data $(u_0,v_0)\in {(C^0(\stackrel{￣}{\mathrm{\Omega }}))}^2$ and $w_0\in W^{1,\infty }(\mathrm{\Omega })$ are non-negative and the parameters $d_1,d_2,d_3>0$, ${\mu }_1,{\mu }_2>0$, $a_1,a_2>0$ and $\alpha ,\beta >0$. The chemotactic function ${\chi }_i(w)$ ($i=1,2$) is smooth and satisfying some conditions. It is proved that the corresponding initial–boundary value problem possesses a unique global bounded classical solution if one of the following cases hold: for $i=1,2$,(i) ${\chi }_i(w)={\chi }_{0,i}>0$ and

${6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}<\frac{\pi \sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}?\frac{2\sqrt{d_i{d}_3}}{\sqrt{n+1}{\chi }_{0,i}}\arctan ?\frac{d_i?d_3}{2}\sqrt{\frac{n+1}{d_i{d}_3}};$

(ii) $0<{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}\le \frac{d_3}{3(n+1){6{\chi }_{i}6}_{L^{\infty }[0,{6w_{0}6}_{L^{\infty }(\mathrm{\Omega })}]}}\min ?\{\frac{2d_i}{d_i+d_3},1\}$.Moreover, we prove asymptotic stabilization of solutions in the sense that:? If $a_1,a_2\in (0,1)$ and $u_0\ne 0\ne v_0$, then any global bounded solution exponentially converge to $(\frac{1?a_1}{1?a_1{a}_2},\frac{1?a_2}{1?a_1{a}_2},0)$ as $t\to \infty$;? If $a_1>1>a_2>0$ and $v_0\ne 0$, then any global bounded solution exponentially converge to $(0,1,0)$ as $t\to \infty$;? If $a_1=1>a_2>0$ and $v_0\ne 0$, then any global bounded solution algebraically converge to $(0,1,0)$ as $t\to \infty$.     

Boundedness vs. blow-up in a chemotaxis system

We determine the critical blow-up exponent for a Keller-Segel-type chemotaxis model, where the chemotactic sensitivity equals some nonlinear function of the particle density. Assuming some growth conditions for the chemotactic sensitivity function we establish an a priori estimate for the solution of the problem considered and conclude the global existence and boundedness of the solution. Furthermore, we prove the existence of solutions that become unbounded in finite or infinite time in that situation where this a priori estimate fails.     

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 of solutions to the critical fully parabolic quasilinear one‐dimensional Keller–Segel system

In this paper we consider a one‐dimensional fully parabolic quasilinear Keller–Segel system with critical nonlinear diffusion. We show uniform‐in‐time boundedness of solutions, which means, that unlike in higher dimensions, there is no critical mass phenomenon in the case of critical diffusion. To this end we utilize estimates from a well‐known Lyapunov functional and a recently introduced new Lyapunov‐like functional in 3 .     

Global well-posedness and asymptotic stabilization for chemotaxis system with signal-dependent sensitivity

A fully parabolic chemotaxis system

$u_t=\mathrm{\Delta }u?\mathrm{?}?(u\chi (v)\mathrm{?}v),v_t=\mathrm{\Delta }v?v+u,$

in a smooth bounded domain $\mathrm{\Omega }?{\mathbb{R}}^N$, $N\ge 2$ with homogeneous Neumann boundary conditions is considered, where the non-negative chemotactic sensitivity function χ satisfies $\chi (v)\le \mu {(a+v)}^{?k}$, for some $a\ge 0$ and $k\ge 1$. It is shown that a novel type of weight function can be applied to a weighted energy estimate for $k>1$. Consequently, the range of μ for the global existence and uniform boundedness of classical solutions established by Mizukami and Yokota [23] is enlarged. Moreover, under a convexity assumption on Ω, an asymptotic Lyapunov functional is obtained and used to establish the asymptotic stability of spatially homogeneous equilibrium solutions for $k\ge 1$ under a smallness assumption on μ. In particular, when $\chi (v)=\mu /v$ and $N<8$, it is shown that the spatially homogeneous steady state is a global attractor whenever $\mu \le 1/2$.     

Blowup of solutions to a two-chemical substances chemotaxis system in the critical dimension

This paper deals with positive solutions of the fully parabolic system

$\{\begin{array}{cc}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_1{v}_t=\mathrm{\Delta }v?v+w\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty ),\hfill \\ {\tau }_2{w}_t=\mathrm{\Delta }w?w+u\hfill & \mathrm{in}\mathrm{\Omega }\times (0,\infty )\hfill \end{array}$

under mixed boundary conditions (no-flux and Dirichlet conditions) in a smooth bounded convex domain $\mathrm{\Omega }?{\mathbb{R}}^4$ with positive parameters ${\tau }_1,{\tau }_2,\chi >0$ and nonnegative smooth initial data $(u_0,v_0,w_0)$.Global existence and boundedness of solutions were shown if ${6u_{0}6}_{L^1(\mathrm{\Omega })}<{(8\pi )}^2/\chi$ in Fujie–Senba (2017). In the present paper, it is shown that there exist blowup solutions satisfying ${6u_{0}6}_{L^1(\mathrm{\Omega })}>{(8\pi )}^2/\chi$. This result suggests that the system can be regard as a generalization of the Keller–Segel system, which has $8\pi /\chi$-dichotomy. The key ingredients are a Lyapunov functional and quantization properties of stationary solutions of the system in ${\mathbb{R}}^4$.     

Global solutions of a two‐dimensional chemotaxis system with attraction and repulsion rotational flux terms

We consider the attraction–repulsion chemotaxis system with rotational flux terms where is a bounded domain with smooth boundary. Here, S₁ and S₂ are given parameter functions on [0,∞)²×Ω with values in . It is shown that for any choice of suitably regular initial data (u₀,v₀,w₀) fulfilling a smallness condition on the norm of v₀,w₀ in L^∞(Ω), the corresponding initial‐boundary value problem possesses a global bounded classical solution. Copyright © 2016 John Wiley & Sons, Ltd.     

Global existence of solutions to a parabolic-parabolic system for chemotaxis with weak degradation

We study the global existence of solutions to a parabolic-parabolic system for chemotaxis with a logistic source in a two-dimensional domain, where the degradation order of the logistic source is weaker than quadratic. We introduce nonlinear production of a chemoattractant, and show the global existence of solutions under certain relations between the degradation and production orders.     

Absence of collapse in a parabolic chemotaxis system with signal‐dependent sensitivity

We consider the chemotaxis system under homogeneous Neumann boundary conditions in a smooth bounded domain Ω ? ?ⁿ. The chemotactic sensitivity function is assumed to generalize the prototype It is proved that no chemotactic collapse occurs in the sense that for any choice of nonnegative initial data (with some r > n), the corresponding initial‐boundary value problem possesses a unique global solution that is uniformly bounded (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Global well-posedness for nonlocal fractional Keller-Segel systems in critical Besov spaces

In this paper, we study Keller-Segel systems with fractional diffusion and a nonlocal term. We establish the global existence, uniqueness and stability of solutions for systems with small initial data in critical Besov spaces. Our main tools are the L^p−L^q estimates for in Besov spaces and the perturbation of linearization.     

Large time behavior of solutions to a fully parabolic chemotaxis–haptotaxis model in N dimensions

This paper deals with the Neumann problem for a fully parabolic chemotaxis–haptotaxis model of cancer invasion given by

$\{\begin{array}{c}{u}_t=\mathrm{\Delta }u?\chi \mathrm{?}?(u\mathrm{?}v)?\xi \mathrm{?}?(u\mathrm{?}w)+u(a?\mu u^{r?1}?\lambda w),x\in \mathrm{\Omega },t>0,\hfill \\ \tau v_t=\mathrm{\Delta }v?v+u,x\in \mathrm{\Omega },t>0,\hfill \\ w_t=?vw,x\in \mathrm{\Omega },t>0.\hfill \end{array}$

Here, $\mathrm{\Omega }?{\mathbb{R}}^N(N\ge 1)$ is a bounded domain with smooth boundary and $\tau >0,r>1,\lambda \ge 0$, $a\in \mathbb{R}$, $\mu ,\xi$ and χ are positive constants. It is shown that the corresponding initial–boundary value problem possesses a unique global bounded classical solution in the cases $r>2$ or $r=2$, with $\mu >{\mu }^{?}=\frac{{(N?2)}_{+}}{N}(\chi +C_{\beta })C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$ for some positive constants $C_{\beta }$ and $C_{\frac{N}{2}+1}$. Furthermore, the large time behavior of solutions to the problem is also investigated. Specially speaking, when a is appropriately large, the corresponding solution of the system exponentially decays to $({(\frac{a}{\mu })}^{\frac{1}{r?1}},{(\frac{a}{\mu })}^{\frac{1}{r?1}},0)$ if μ is large enough. This result improves or extends previous results of several authors.     

Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis

In this paper, we establish the existence and the nonlinear stability of traveling wave solutions to a system of conservation laws which is transformed, by a change of variable, from the well-known Keller-Segel model describing cell (bacteria) movement toward the concentration gradient of the chemical that is consumed by the cells. We prove the existence of traveling fronts by the phase plane analysis and show the asymptotic nonlinear stability of traveling wave solutions without the smallness assumption on the wave strengths by the method of energy estimates.     

On a quasilinear parabolic–elliptic chemotaxis system with logistic source

This paper deals with a quasilinear parabolic–elliptic chemotaxis system with logistic source, under homogeneous Neumann boundary conditions in a smooth bounded domain. For the case of positive diffusion function, it is shown that the corresponding initial boundary value problem possesses a unique global classical solution which is uniformly bounded. Moreover, if the diffusion function is zero at some point, or a positive diffusion function and the logistic damping effect is rather mild, we proved that the weak solutions are global existence. Finally, it is asserted that the solutions approach constant equilibria in the large time for a specific case of the logistic source.     

Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model

We consider the classical parabolic-parabolic Keller-Segel system