Quasilinear Chemotaxis System Research Articles

Global boundedness and large time behaviour in a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant

This paper deals with the following quasilinear chemotaxis system with consumption of chemoattractant \[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad & x\in \Omega,\quad t>0,\\ v_t=\Delta v-uv,\quad & x\in \Omega,\quad t>0\\ \end{array}\right. \] in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$ with smooth boundary $\partial \Omega$ . It is shown that if $m>\max \{1,\,\frac {3N-2}{2N+2}\}$ , for any reasonably smooth nonnegative initial data, the corresponding no-flux type initial-boundary value problem possesses a globally bounded weak solution. Furthermore, we prove that the solution converges to the spatially homogeneous equilibrium $(\bar {u}_0,\,0)$ in an appropriate sense as $t\rightarrow \infty$ , where $\bar {u}_0=\frac {1}{|\Omega |}\int _\Omega u_0$ . This result not only partly extends the previous global boundedness result in Fan and Jin (J. Math. Phys.58 (2017), 011503) and Wang and Xiang (Z. Angew. Math. Phys.66 (2015), 3159–3179) to $m>\frac {3N-2}{2N}$ in the case $N\geq 3$ , but also partly improves the global existence result in Zheng and Wang (Discrete Contin. Dyn. Syst. Ser. B22 (2017), 669–686) to $m>\frac {3N}{2N+2}$ when $N\geq 2$ .

Blow-up Analysis to a Quasilinear Chemotaxis System with Nonlocal Logistic Effect

Blow-up Analysis to a Quasilinear Chemotaxis System with Nonlocal Logistic Effect

Boundedness in a three-component quasilinear chemotaxis system on Alopecia Areata

Boundedness in a three-component quasilinear chemotaxis system on Alopecia Areata

Boundedness and asymptotic behavior in a quasilinear chemotaxis system for alopecia areata

In this paper, we study the following chemotaxis system with generalized volume-filling effect ut=∇⋅(D1(u)∇u)−χ1∇⋅(S1(u)∇w)+w−μ1uη1,(x,t)∈Ω×(0,∞),vt=∇⋅(D2(v)∇v)−χ2∇⋅(S2(v)∇w)+w+ruv−μ2vη2,(x,t)∈Ω×(0,∞),0=Δw+u+v−w,(x,t)∈Ω×(0,∞),under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂Rn(n≥1), which describes the spatio-temporal dynamics of alopecia areata lesions, where χ1, χ2, r, μ1, μ2 are positive parameters and η1,η2≥2. When the functions Di and Si(i=1,2) belong to C2 fulfilling Di(s)≥(s+1)αi, 0≤S(s)≤sβi with αi∈R and βi≥0 for all s≥0, we study the global existence and boundedness of classical solutions for the above system under some suitable conditions, and find that either the higher-order nonlinear diffusion or strong logistic damping can prevent blow-up of classical solutions for the problem. In addition, when ηi=2 and 0≤Si(s)≤s(s+1)βi−1 with 0≤βi<1 or 0≤Si(s)≤sβi with βi≥1 for all s≥0 and i=1,2, if the chemosensitivity χ1 and χ2 are appropriately mild and some other parameter conditions hold, the globally bounded solution will stabilize to the constant coexistence equilibrium as the time goes to infinity. Our results not only extend the previous ones of Tao & Xu (2022), but also involve some new conclusions.

Global Boundedness of Solutions to a Quasilinear Chemotaxis System with Nonlocal Nonlinear Reaction

Global Boundedness of Solutions to a Quasilinear Chemotaxis System with Nonlocal Nonlinear Reaction

Boundedness and Long-Time Behavior for a Two-Dimensional Quasilinear Chemotaxis System with Indirect Signal Consumption

Boundedness and Long-Time Behavior for a Two-Dimensional Quasilinear Chemotaxis System with Indirect Signal Consumption

A new result for boundedness of solutions to a higher-dimensional quasilinear chemotaxis system with a logistic source

We deal with the boundedness of solutions of a class of quasilinear chemotaxis systems generalizing the prototype(⁎){ut=∇⋅(ϕ(u)∇u)−∇⋅(u∇v)+μu(1−u),x∈Ω,t>0,vt=Δv−v+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω, where RN(N≥2) is a bounded domain with zero-flux boundary condition, μ>0 is a positive parameter, and ϕ(u)=(u+1)−α. It is shown that for all reasonably regular initial data, system (⁎) admits a global existence and boundedness of solutions when α<α0 for some α0>2−NN+2, thereby proving that the value α=2−NN+2 is not critical in this regard. This extends previous results that rely on a different energy-type inequality.

Boundedness of the Higher-Dimensional Quasilinear Chemotaxis System with Generalized Logistic Source

This article considers the following higher-dimensional quasilinear parabolic-parabolic-ODE chemotaxis system with generalized Logistic source and homogeneous Neumann boundary conditions $$\begin{cases}u_t=\triangledown\cdot(D(u)\triangledown{u})-\triangledown\cdot(S(u)\triangledown{v})+f(u), & x \in \Omega, t > 0\\v_t=\Delta{v}+w-v, & x \in \Omega, t > 0,\\w_t=u-w, & x \in \Omega, t > 0,\end{cases}$$ in a bounded domain Ω ⊂ Rn(n ≥ 2) with smooth boundary ∂Ω, where the diffusion coefficient D(u) and the chemotactic sensitivity function S(u) are supposed to satisfy D(u) ≥ M1(u + 1)−α and S(u) ≤ M2(u + 1)β, respectively, where M1, M2 > 0 and α, β ∈ R. Moreover, the logistic source f(u) is supposed to satisfy f(u) ≤ a — µuγ with µ > 0, γ ≥ 1, and a ≥ 0. As $$\alpha+2\beta<\gamma-1+\frac{2\gamma}{n}$$ , we show that the solution of the above chemotaxis system with sufficiently smooth nonnegative initial data is uniformly bounded.

Asymptotic behavior in a quasilinear chemotaxis-growth system with indirect signal production

We consider a quasilinear chemotaxis system involving logistic source{ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v)+μ(u−uγ),x∈Ω,t>0,vt=Δv−v+w,x∈Ω,t>0,wt=Δw−w+u,x∈Ω,t>0, with nonnegative initial data under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n⩾1). Here, constants μ>0, γ>1, and D, S are smooth functions fulfilling D(s)⩾K0(s+1)α, |S(s)|⩽K1s(s+1)β−1 for all s⩾0 with α,β∈R and K0,K1>0. Then, if β⩽γ−1, the nonnegative classical solution (u,v,w) is global in time and bounded. Moreover, if μ>0 is sufficiently large, this global bounded solution with nonnegative initial data (u0,v0,w0) satisfies‖u(⋅,t)−1‖L∞(Ω)+‖v(⋅,t)−1‖L∞(Ω)+‖w(⋅,t)−1‖L∞(Ω)→0 as t→∞.

Large time behavior of solution to quasilinear chemotaxis system with logistic source

This paper deals with the quasilinear parabolic-elliptic chemotaxis system \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} u_{t} = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(\chi u \nabla v)+\mu u- \mu u^{r}, \, \, \, x'>under homogeneous Neumann boundary conditions in a bounded domain \begin{document}$ \Omega\subset\mathbb{R}^{n} $\end{document} with smooth boundary, where \begin{document}$ \tau\in\{0, 1\} $\end{document} , \begin{document}$ \chi>0 $\end{document} , \begin{document}$ \mu>0 $\end{document} and \begin{document}$ r\geq2 $\end{document} . \begin{document}$ D(u) $\end{document} is supposed to satisfy \begin{document}$ \begin{equation*} \begin{split} D(u)\geq (u+1)^{\alpha} \, \, \, \text{with}\, \, \, \alpha>0. \end{split} \end{equation*} $\end{document} It is shown that when \begin{document}$ \mu>\frac{\chi^{2}}{16} $\end{document} and \begin{document}$ r\geq2 $\end{document} , then the solution to the system exponentially converges to the constant stationary solution \begin{document}$ (1, 1) $\end{document} .

Finite-time blow-up in quasilinear parabolic-elliptic chemotaxis system with nonlinear signal production

In this paper, a quasilinear chemotaxis system of parabolic-elliptic type(0.1){ut=∇⋅(φ(u)∇u)−∇⋅(u∇v)x∈Ω,t>0,0=Δv−μ(t)+f(u),μ(t):=1|Ω|∫Ωf(u(⋅,t)),x∈Ω,t>0 is considered associated with homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn, n≥1. The diffusivity φ(ξ)∈C2([0,∞)) is a positive function and we particularly suppose that φ(ξ)=C0(1+ξ)−m for all ξ≥0 with C0>0 and m∈R. f is a suitably regular positive function given by f(ξ)=K(1+ξ)κ for all ξ≥0 with parameters K>0 and κ>0. It is shown that there exist regular initial data u0 such that•if κ+m<2n, then all solutions of (0.1) are global and bounded;•if κ+m>2n, however, there exists nonnegative radially symmetric initial data u0 such that the corresponding solution of (0.1) blows up in finite time. The critical exponent 2n seems to be essentially optimal which was given by [30].

Global existence and uniqueness of solutions to a chemotaxis system

ABSTRACT This paper is dealing with the existence and uniqueness of a global in-time bounded solution to a quasilinear chemotaxis system settled in a bounded domain under no-flux boundary conditions. The main difficulty occurs in the presence of the nonlinear logarithmic diffusion which can blow up around zero, and the energy estimates which do not provide a uniform estimate for large values of cell density u. Moreover, the chemo-sensitivity function depends on both cell and chemo-attractant densities which presents a further difficulty while proving the global existence of a solution. This is overcome by tracking the time evolution of a suitable functional. The existence of local in-time weak solutions is ensured using Schauder's fixed point theorem, while the uniqueness is obtained by adapting the method introduced in Diaz et al. [On a quasilinear degenerate system arising in semiconductor theory. Part I: existence and uniqueness of solutions. Nonlinear Anal. Real World Appl. 2001;2:305–336.]. Furthermore, under appropriate assumptions on the initial data, we prove that the solution is classical by using parabolic regularity results.

Finite-time blow-up in a quasilinear chemotaxis system with an external signal consumption

This paper deals with a quasilinear chemotaxis system with an external signal consumption \begin{equation*}%\label{1a} \begin{cases} u_t=\nabla\cdot(\varphi(u)\nabla u)-\nabla\cdot(u\nabla v), &amp;(x,t)\in \Omega\times (0,\infty), \\ 0=\Delta v+u-g(x), &amp;(x,t)\in \Omega\times (0,\infty), \end{cases} \end{equation*} under homogeneous Neumann boundary conditions in a ball $\Omega\subset \mathbb{R}^{n}$, where $\varphi(u)$ is a nonlinear diffusion function and $g(x)$ is an external signal consumption. Under suitable assumptions on the functions $\varphi$ and $g$, it is proved that there exists initial data such that the solution of the above system blows up in finite time.

Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant

Boundedness in a quasilinear two-species chemotaxis system with consumption of chemoattractant

Global existence of bounded solutions for a quasilinear chemotaxis system with logistic source

In this paper, we deal with the chemotaxis system with logistic source under homogeneous Neumann boundary conditions in a bounded convex domain with smooth boundary. Under appropriate regularity assumptions on the initial data, by developing some Lp-estimate techniques, we show that the system possesses at least one global and bounded weak solution. Our results generalized and improved previous results, and partially results are new.

Boundedness and large time behavior in a quasilinear chemotaxis model for tumor invasion

This paper deals with the quasilinear chemotaxis system modeling tumor invasion [Formula: see text] under homogenous Neumann boundary conditions in a smoothly convex bounded domain [Formula: see text] [Formula: see text], where [Formula: see text] is a given function satisfying [Formula: see text] for all [Formula: see text] with [Formula: see text] and [Formula: see text]. Here the matrix-valued function [Formula: see text] fulfills [Formula: see text] for all [Formula: see text] with some [Formula: see text] and [Formula: see text]. It is shown that for all reasonably regular initial data, a corresponding initial-boundary value problem for this system possesses a globally defined weak solution under some assumptions. Based on this boundedness property, it can finally be proved that in the large time limit, any such solution approaches the spatially homogenous equilibrium [Formula: see text] in an appropriate sense, where [Formula: see text], [Formula: see text] and [Formula: see text] provided that merely [Formula: see text] on [Formula: see text]. To the best of our knowledge, there are the first results on boundedness and asymptotic behavior of the system.

Global dynamics of a quasilinear chemotaxis model arising from tumor invasion

This paper is concerned with a quasilinear chemotaxis system ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),x∈Ω,t>0,vt=Δv+wz,x∈Ω,t>0,wt=−wz,x∈Ω,t>0,zt=Δz−z+u,x∈Ω,t>0,with homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n≥1), where D satisfies D(u)>0 for all u≥0 and behaves algebraically as u→∞. It is shown that if S(u)D(u)≤Cuα with some constants C>0 for all u≥1 and α<1+1n,if1≤n≤3,α<4n,ifn≥4,then for sufficiently smooth initial data, the system possesses a unique bounded classical solution (u,v,w,z), which exponentially converges to the equilibrium (ū0,v̄0+w̄0,0,ū0) as t→+∞, where ū0=1|Ω|∫Ωu0(x)dx, v̄0=1|Ω|∫Ωv0(x)dx andw̄0=1|Ω|∫Ωw0(x)dx.

A blow-up result for a quasilinear chemotaxis system with logistic source in higher dimensions

In this paper we consider the quasilinear chemotaxis system{ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+f(u),x∈Ω,t>0,0=Δv−μ(t)+u,x∈Ω,t>0, with homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rn with n≥2, where χ>0, μ(t):=1|Ω|∫Ωu(x,t)dx and f∈C([0,∞))∩C1((0,∞)) is a logistic source of the form f(s)=as−bsκ with a≥0,b>0, κ>1 and s≥0, and the diffusion D∈C2([0,∞)) is supposed to satisfyD(s)≥D0s−mfor alls>0 with some D0>0 and m∈R. Given any b>0, when the logistic source is strong enough in the sense thatκ>m+3−4n+2andκ>2, it is shown that for any initial data u0∈C0(Ω¯) and n≥2 the problem possesses a unique global bounded classical solution. However, whenD(s)=D0s−mfor alls>0 with 4n−1<m≤0 in the sense that n≥5, and the effect of logistic source is weaker in the sense thatκ∈(1,(3−m)n−22n−2), it is shown that for arbitrary prescribed M0>0 there exists initial data u0∈C∞(Ω¯) satisfying ∫Ωu0=M0 such that the corresponding solution (u,v) of the system blows up in finite time in a ball Ω=B0(R)⊂Rn with some R>0. This result extends the blow-up arguments of the Keller–Segel chemotaxis model with logistic cell kinetics in Winkler [39] to more general quasilinear case. Moreover, since there is a gap in the proof of Zheng et al. [46], it also presents modified results for the mistake.

Global boundedness in a fully parabolic quasilinear chemotaxis system with singular sensitivity

In this paper we study the global boundedness of solutions to the quasilinear fully parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇φ(v)), vt=Δv−v+u, where bounded domain Ω⊂Rn (n≥2) subject to the non-flux boundary conditions, the diffusivity fulfills D(u)=a0(u+1)−α with a0>0 and α≥0, while the density-signal governed sensitivity satisfies 0≤S(u)≤b0(u+1)β and 0<φ′(v)≤χvk for b0,χ>0 and β,k∈R. It is shown that the solution is globally bounded provided α+β<1 and k≤1. This result demonstrates the effect of signal-dependent sensitivity on the blow-up prevention.

Global boundedness in a quasilinear chemotaxis system with general density-signal governed sensitivity

In this paper we study the global boundedness of solutions to the quasilinear parabolic chemotaxis system: ut=∇⋅(D(u)∇u−S(u)∇φ(v)), 0=Δv−v+u, subject to homogeneous Neumann boundary conditions and the initial data u0 in a bounded and smooth domain Ω⊂Rn (n≥2), where the diffusivity D(u) is supposed to satisfy D(u)≥a0(u+1)−α with a0>0 and α∈R, while the density-signal governed sensitivity fulfills 0≤S(u)≤b0(u+1)β and 0<φ′(v)≤χvk for b0,χ>0 and β,k∈R. It is shown that the solution is globally bounded if α+β<(1−2n)k+2n with n≥3 and k<1, or α+β<1 for k≥1. This implies that the large k benefits the global boundedness of solutions due to the weaker chemotactic migration of the signal-dependent sensitivity at high signal concentrations. Moreover, when α+β arrives at the critical value, we establish the global boundedness of solutions for the coefficient χ properly small. It should be emphasized that the smallness of χ under k>1 is positively related to the total cellular mass ∫Ωu0dx, which is attributed to the stronger singularity of φ(v) at v=0 for k>1 and the fact that v can be estimated from below by a multiple of ∫Ωu0dx. In addition, distinctive phenomena concerning this model are observed by comparison with the known results.