Quasilinear Chemotaxis Research Articles

Threshold value for a quasilinear Keller–Segel chemotaxis system with the intermediate exponent in a bounded domain

We consider a quasilinear chemotaxis model ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),τvt=Δv−v+u, with nonlinear diffusion function D∈C2([0,∞)) and chemotactic sensitivity S∈C2([0,∞)) in a bounded domain Ω⊂Rd(d≥3). Here the rate D(s)/S(s) grows like s2−m with 2d/(d+2)<m<2−2/d as s→∞, and τ=0,1.It is first shown that there exists a M∗>0 such that if free energy with initial data is suitably small and ‖u0‖L1(Ω)α‖u0‖Lm(Ω)β<M∗ with α=2/(2−m)−d/m>0 and β=d−2/(2−m)>0, then the classical solutions to the above system are uniformly-in-time bounded. Second, in radially symmetric settings we can find M∗>0 such that ‖u0‖L1(Ω)α‖u0‖Lm(Ω)β>M∗ and the corresponding solution must be unbounded. These results show that the global behavior of classical solutions is classified by the combination of norms of initial data when 2d/(d+2)<m<2−2/d.

Global boundedness in a quasilinear chemotaxis–consumption system with degenerate signal-dependent motility and logistic source

Global boundedness in a quasilinear chemotaxis–consumption system with degenerate signal-dependent motility and logistic source

No Pattern Formation in a Quasilinear Chemotaxis Model with Local Sensing

No Pattern Formation in a Quasilinear Chemotaxis Model with Local Sensing

Large time behavior of solution to a quasilinear chemotaxis model describing tumor angiogenesis with/without logistic source

In this paper, we deal with the following Neumann-initial boundary value problem for a quasilinear chemotaxis model describing tumor angiogenesis: ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+μu−μu2,x∈Ω,t>0,vt=Δv+∇⋅(v∇w)−v+u,x∈Ω,t>0,0=Δw−w+u,x∈Ω,t>0,∂u∂ν=∂v∂ν=∂w∂ν=0,x∈∂Ω,t>0,u(x,0)=u0(x),v(x,0)=v0(x),x∈Ω,in a bounded smooth domain Ω⊂Rn(n≤3), where the parameter χ,ξ>0,μ≥0, D(u) is supposed to satisfy the behind property D(u)≥(u+1)αwithα>0.Assume that either μ≥0,α>1 or μ=0,ξ≥λ1∗χ2, where the parameter λ1∗=λ1∗(u0,v0,Ω)>0, then the system admits a global classical solution (u,v,w) by subtle energy estimates. Moreover, when μ=0, it is asserted that the corresponding solution exponentially converges to the constant stationary solution (u0¯,u0¯,u0¯) provided the initial data u0 is sufficiently small, where u0¯=∫Ωu0|Ω|. Finally, when μ>0, it can be proved that the corresponding solution of the system decays to (1,1,1) exponentially for suitable large μ.

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 \[ \left\{\begin{array}{@{}ll} u_t=\Delta u^{m}-\nabla\cdot(u\nabla v),\quad &amp; x\in \Omega,\quad t&gt;0,\\ v_t=\Delta v-uv,\quad &amp; x\in \Omega,\quad t&gt;0\\ \end{array}\right. \] in a bounded domain $\Omega \subset \mathbb {R}^N(N=3,\,4,\,5)$ with smooth boundary $\partial \Omega$ . It is shown that if $m&gt;\max \{1,\,\frac {3N-2}{2N+2}\}$ , 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 $m&gt;\frac {3N-2}{2N}$ 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 $m&gt;\frac {3N}{2N+2}$ 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

Global existence of a quasilinear chemotaxis model with signal-dependent motility and indirect signal production mechanism

In this paper, we study the following quasilinear chemotaxis model with signal-dependent motility: nt = Δ(γ(c)nm); ct = dcΔc − c + v; vt = dvΔv − v + n, x ∈ Ω, t &amp;gt; 0, ∂(nmγ(c))∂ν=∂c∂ν=∂v∂ν=0, x ∈ ∂Ω, t &amp;gt; 0, n(x, 0) = n0(x), c(x, 0) = c0(x), v(x, 0) = v0(x), x ∈ Ω, t &amp;gt; 0, where γ(c) = c−r. We show that the above system admits at least one global weak solution.

Large time behavior in a quasilinear chemotaxis model with indirect signal absorption

This paper considers the following chemotaxis system ut=∇⋅D(u)∇u−∇⋅S(u)∇v,x∈Ω,t>0,vt=Δv−vw,x∈Ω,t>0,wt=−δw+u,x∈Ω,t>0,under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂Rn(n⩾2), where the parameter δ>0 and D, S are smooth functions satisfying D(u)⩾K0(u+1)α, 0⩽S(u)⩽K1(u+1)β−1u with α,β∈R and K0,K1>0. Suppose that β−α<1n+12, this system admits a global bounded classical solution (u,v,w) fulfilling ‖u⋅t−ū0‖L∞Ω+‖v⋅t−0‖L∞Ω+‖w⋅t−ū0δ‖L∞Ω→0as t→∞, where ū0≔1|Ω|∫Ωu0.

Boundedness of Solutions in a Fully Parabolic Quasilinear Chemotaxis Model with Two Species and Two Chemicals

Boundedness of Solutions in a Fully Parabolic Quasilinear Chemotaxis Model with Two Species and Two Chemicals

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

Boundedness in a Quasilinear Chemotaxis Model with Logistic Growth and Indirect Signal Production

Boundedness in a Quasilinear Chemotaxis Model with Logistic Growth and Indirect Signal Production

Boundedness to a quasilinear chemotaxis–consumption system with singular sensitivity in dimension one

Boundedness to a quasilinear chemotaxis–consumption system with singular sensitivity in dimension one

Global boundedness of solutions to the quasilinear chemotaxis model with flux limitation

Abstract Funded by 国家自然科学基金(11771062,11971082,11601053和11526042) 中央高校基本科研业务费(2020CDJQY-Z001,XDJK2020C054和 2019CDJCYJ001) 中国博士后科学基金(2020M673102) 重庆市自然科学基金(cstc2019jcyj-msxmX0082,cstc2020jcyj-bshX0071) 重庆市博士后创新人才支持计划和重庆市分析数学与应用重点实验室 chemotaxis flux limitation global boundedness × Update Abstract English Chinese Update

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.

A new (and optimal) result for the boundedness of a solution of a quasilinear chemotaxis–haptotaxis model (with a logistic source)

This article deals with an initial-boundary value problem for the coupled chemotaxis–haptotaxis system with nonlinear diffusion(0.1){ut=∇⋅(D(u)∇u)−χ∇⋅(u∇v)−ξ∇⋅(u∇w)+μu(1−u−w),x∈Ω,t>0,τvt=Δv−v+u,x∈Ω,t>0,wt=−vw,x∈Ω,t>0 under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂RN(N≥1), where τ∈{0,1} and χ, ξ, and μ are given nonnegative parameters. The diffusivity D(u) is assumed to satisfyD(u)≥CD(u+1)m−1for allu≥0andCD>0. In the present work it is shown that ifm≥2−2Nλwith0<μ<κ0,m>2−2Nλwithμ≥κ0, orm>2−2Nandμ=0 orm=2−2NandCD>CGN(1+‖u0‖L1(Ω))34(2−2N)2κ0, then for all reasonably regular initial data, a corresponding initial-boundary value problem for (0.1) possesses a unique global classical solution that is uniformly bounded in Ω×(0,∞), whereλ=κ0(κ0−μ)+ andκ0={maxs≥1⁡λ01s+1(χ+ξ‖w0‖L∞(Ω))ifτ=1,χifτ=0. Here CGN and λ0 are constants that correspond to the Gagliardo–Nirenberg inequality and the maximal Sobolev regularity, respectively. With use of new Lp-estimate techniques to obtain the a priori estimate of a solution from L1(Ω)→Lλ−ε(Ω)→Lλ(Ω)→Lλ+ε(Ω)→Lp(Ω)(for allp>1), these results significantly improve or extend previous results obtained by several authors.

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→∞.