Attraction Repulsion Chemotaxis System Research Articles

Global weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source

Global weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and logistic source

Finite-time blow-up and boundedness in a quasilinear attraction–repulsion chemotaxis system with nonlinear signal productions

This article is focused on the no-flux attraction–repulsion chemotaxis model ut=∇⋅(D(u)∇u−S(u)∇v+T(u)∇w)+h(u),x∈Ω,t>0,0=Δv−α(t)+f(u),x∈Ω,t>0,0=Δw−β(t)+g(u),x∈Ω,t>0defined in a smooth and bounded domain Ω⊂Rn(n≥2) and subjected to homogeneous Neumann boundary conditions. The functions D,S and T suitably generalize the singular prototypes D(s)=(s+1)m1−1,S(s)=s(s+1)m2−1,T(s)=s(s+1)m3−1,s≥0,m1,m2,m3∈R,and f and g extend the prototypes f(s)=(s+1)γ1,g(s)=(s+1)γ2,s≥0,γ1,γ2>0,as well as the logistic dampening satisfies h(s)=ks−μsl with k∈R, μ>0, l≥1. Under alternative conditions, we establish the global existence and boundedness of classical solutions to the above model. However, when m2=m3 and no logistic source is taken into account, the finite time blow-up phenomenon preserves provided that γ1>γ2 and γ1>m1∗−m2+2n with m1∗=max{1,m1}.

Finite-time blow-up for the attractive dominant case of an attraction-repulsion chemotaxis system in the whole space

Finite-time blow-up for the attractive dominant case of an attraction-repulsion chemotaxis system in the whole space

Global boundedness in an attraction-repulsion chemotaxis system involving nonlinear indirect signal mechanism

We study the following chemotaxis system involving nonlinear indirect signal mechanism{ut=Δu−ξ∇⋅(u∇v)+χ∇⋅(u∇z),x∈Ω,t>0,vt=Δv−v+wγ1,x∈Ω,t>0,0=Δw−w+uγ2,x∈Ω,t>0,0=Δz−z+uγ3,x∈Ω, under homogeneous Neumann boundary conditions in a bounded and smooth domain Ω⊂Rn(n≥1), where ξ,χ,γ1,γ2,γ3>0. With the aid of maximum Sobolev regularity and a priori estimates of w and z, it has been proved that if either random diffusion or repulsion mechanism dominates the nonlinear indirect attraction mechanism with γ1γ2<max⁡{2n,γ3}, then the system admits a global classical solution, which is bounded in Ω×(0,∞). Moreover, we also show that under balance situations with γ1γ2=max⁡{2n,γ3}, the existence of global classical solution is determined by the sizes of chemotaxis sensitivity coefficients ξ and χ as well as the initial data u0. This work extends the previous results, where the production of signal is nonlinear mechanism.

Global boundedness in a two-species attraction–repulsion chemotaxis system with two chemicals and nonlinear productions

This paper studies the attraction–repulsion chemotaxis system of two-species with two chemicals ut=Δu−χ1∇⋅(u∇¯v)+f1(u), vt=Δv−v+wγ1, wt=Δw+χ2∇⋅(w∇z)+f2(w), 0=Δz−z+uγ2, subject to the homogeneous Neumann boundary conditions in a bounded domain Ω⊂RN (N≥2) with smooth boundary, where the parameters χi,γi>0(i=1,2), and the logistic sources fi(s)∈C2[0,∞) satisfy fi(s)≤μis(1−sθi) with μi,θi>0, fi(0)≥0 (i=1,2). The interactions among the diffusion, attraction, repulsion, logistic sources, and nonlinear productions in the system determine the behavior of solutions. It is obtained that the solutions would be globally bounded whenever the nonlinear productions are dominated by one of the following three mechanisms: (i) the diffusion with γ1<2N, or γ2<4N with γ2≤1; (ii) the logistic sources with min{θ1,θ2}≥max{γ1,γ2}, and (iii) the cooperation of diffusion and logistic sources with θ1+1>γ2min{1+N2,1+Nγ12θ2}, or θ2+1>γ1min{1+N2,1+max{N4,1}γ2θ1}.

Global existence of solutions to the 4D attraction–repulsion chemotaxis system and applications of Brezis–Merle inequality

We consider the Cauchy problem for an attraction–repulsion chemotaxis system in the four space dimension. One of main topics in the study of such a system is the presence of L 1 threshold. In fact, the critical mass phenomenon called 8π-problem is well-known in the two-dimensional setting. In this paper, we show the global existence of solutions to the Cauchy problem for the four-dimensional subcritical case, that is, the total mass of the initial data is less than the four-dimensional L 1 threshold value . The key ingredients are the four-dimensional Brezis–Merle type inequality of the 4th-order elliptic equation and some inequalities derived from rearrangement arguments.

On a Two-Species Attraction–Repulsion Chemotaxis System with Nonlocal Terms

On a Two-Species Attraction–Repulsion Chemotaxis System with Nonlocal Terms

Refined criteria toward boundedness in an attraction–repulsion chemotaxis system with nonlinear productions

We study some zero-flux attraction-repulsion chemotaxis models, with nonlinear production rates for the chemorepellent and the chemoattractant, whose formulation can be schematized as (⋄) In this problem, Ω is a bounded and smooth domain of , for , , , reasonably regular functions generalizing, respectively, the prototypes and , for some and all . Moreover, and have specific expressions, and . Once for any sufficiently smooth , and the local well-posedness of problem () is ensured, and we establish for the classical solution defined in that the life span is indeed and u, v and w are uniformly bounded in in the following cases: For , , , and , provided (I.1) k<l; (I.2) ; (I.3) k = l and , or and . For , , , and , whenever (II.1) ; (II.2) and , or and ; (II.3) . For and and , under the assumptions k<l or (I.3)). In particular, in this paper we partially improve what derived in Viglialoro [Influence of nonlinear production on the global solvability of an attraction-repulsion chemotaxis system. Math Nachr. 2021;294(12):2441–2454] and solve an open question given in Liu and Li [Finite-time blowup in attraction-repulsion systems with nonlinear signal production. Nonlinear Anal Real World Appl. 2021;61:Paper No. 103305, 21]. Finally, the research is complemented with numerical simulations in bi-dimensional domains.

Spreading speed in a fractional attraction–repulsion chemotaxis system with logistic source

This paper studies an attraction-repulsion chemotaxis system with logistic source and a fractional diffusion of order α∈(0,1) on RN: (0.1)ut=−(−Δ)αu−χ1∇⋅(u∇v1)+χ2∇⋅(u∇v2)+u(a−bu),x∈RN,t>0,0=(Δ−λ1I)v1+μ1u,x∈RN,t>0,0=(Δ−λ2I)v2+μ2u,x∈RN,t>0,where α, χi, μi, λi(i=1,2), a and b are constants. We show that (0.1) has a unique global bounded solution and investigate the asymptotic behavior of the global classical solutions under some assumptions on the parameters. Particularly, we consider the spreading properties of the global classical solutions. Under some conditions for the nonnegative initial function u0, we prove that (0.2)lim inft→∞inf|x|≤ectu(x,t;u0)>0,for all0<c<aN+2αand (0.3)limt→∞sup|x|≥ectu(x,t;u0)=0,for allc>aN+2α.This fact shows that the spreading speed of (0.1) is exponential in time, which is different from the linear spreading speed obtained in Salako and Shen (2019).

Global boundedness in a parabolic-parabolic-elliptic attraction-repulsion chemotaxis system with nonlinear productions

This paper investigates the global boundedness of the semilinear attraction-repulsion chemotaxis system with nonlinear productions and logistic source: \begin{document}$ \begin{cases} u_{t} = \Delta u-\chi\nabla\cdot(u\nabla v)+\xi\nabla\cdot(u\nabla w)+u(a-bu^s), \ (x, t)\in \Omega\times (0, T), \\[6pt] v_t = \Delta v+\alpha u^k-\beta v, \ (x, t)\in \Omega\times (0, T), \\[6pt] 0 = \Delta w+\gamma u^l-\delta w, \ (x, t)\in\Omega\times (0, T), \end{cases} $\end{document} with $ \alpha, \beta, \gamma, \delta, \chi, \xi, a, b, s, k, l&gt;0 $, subject to the homogeneous Neumann boundary conditions in a bounded domain $ \Omega\subset\mathbb{R}^n $ ($ n\ge1 $) with smooth boundary. Hong et al. (J. Math. Anal. Appl. 484: 12703, 2020) have established the global boundedness criteria for the parabolic-elliptic-elliptic attraction-repulsion chemotaxis system. With the help of the maximum Sobolev regularity, this paper generalizes the results to the parabolic-parabolic-elliptic case. It is proved that if one of the random diffusion, logistic source and repulsion mechanisms dominates the attraction with $ \max\{l, s, \frac{2}{n}\}&gt;k $, the solutions would be globally bounded. Under the balance situations, the boundedness of solutions will be determined by the sizes of the coefficients involved, but Hong et al only considered the balance of logistic source and repulsion mechanisms with $ \max\{l, s\} = k $, the balance produced by random diffusion was not taken into account, this paper consider the balance of random diffusion, logistic source and repulsion mechanisms with $ \max\{l, s, \frac{2}{n}\} = k $, the clear characterization of the global boundedness criteria is obtained for the parabolic-parabolic-elliptic attraction-repulsion chemotaxis system.

Large time behavior in a quasilinear parabolic-parabolic-elliptic attraction-repulsion chemotaxis system

This paper deals with a quasilinear parabolic-parabolic-elliptic attraction-repulsion chemotaxis system. Boundedness, stabilization and blow-up in this system of the fully parabolic and parabolic-elliptic-elliptic versions have already been proved. The purpose of this paper is to derive boundedness and stabilization in the parabolic-parabolic-elliptic version.

Global boundedness in a quasilinear two-species attraction-repulsion chemotaxis system with two chemicals

&lt;p style='text-indent:20px;'&gt;This paper studies the quasilinear attraction-repulsion chemotaxis system of two-species with two chemicals &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ u_{t} = \nabla\cdot( D_1(u)\nabla u)-\nabla\cdot( \Phi_1(u)\nabla v) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 0 = \Delta v-v+w^{\gamma_1} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ w_{t} = \nabla\cdot( D_2(w)\nabla w)+\nabla\cdot( \Phi_2(w)\nabla z) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ 0 = \Delta z-z+u^{\gamma_2} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, subject to the homogeneous Neumann boundary conditions in a bounded domain &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ \Omega\subset\mathbb{R}^N $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;(&lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ N\geq2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) with smooth boundary, where &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;\begin{document}$ \gamma_i&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M8"&gt;\begin{document}$ D_i,\Phi_i\in C^2[0,+\infty) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M9"&gt;\begin{document}$ D_i(s)\ge(s+1)^{p_i},\; \Phi_i(s)\ge0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for &lt;inline-formula&gt;&lt;tex-math id="M10"&gt;\begin{document}$ s\ge 0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, and &lt;inline-formula&gt;&lt;tex-math id="M11"&gt;\begin{document}$ \Phi_i(s)\le\chi_i s^{q_i} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for &lt;inline-formula&gt;&lt;tex-math id="M12"&gt;\begin{document}$ s&amp;gt;s_0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with &lt;inline-formula&gt;&lt;tex-math id="M13"&gt;\begin{document}$ \chi_i&amp;gt;0 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M14"&gt;\begin{document}$ p_i,q_i\in\mathbb{R} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; &lt;inline-formula&gt;&lt;tex-math id="M15"&gt;\begin{document}$ (i = 1,2) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M16"&gt;\begin{document}$ s_0&amp;gt;1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. It is shown that if &lt;inline-formula&gt;&lt;tex-math id="M17"&gt;\begin{document}$ \gamma_1&amp;lt;\frac{2}{N} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (or &lt;inline-formula&gt;&lt;tex-math id="M18"&gt;\begin{document}$ \gamma_2&amp;lt;\frac{4}{N} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; with &lt;inline-formula&gt;&lt;tex-math id="M19"&gt;\begin{document}$ \gamma_2\le1 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;), the global boundedness of solutions are guaranteed by the self-diffusion dominance of &lt;inline-formula&gt;&lt;tex-math id="M20"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (or &lt;inline-formula&gt;&lt;tex-math id="M21"&gt;\begin{document}$ w $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;) with &lt;inline-formula&gt;&lt;tex-math id="M22"&gt;\begin{document}$ p_1&amp;gt;q_1+\gamma_1-1-\frac{2}{N} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (or &lt;inline-formula&gt;&lt;tex-math id="M23"&gt;\begin{document}$ p_2&amp;gt;q_2+\gamma_2-1-\frac{4}{N} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;); if &lt;inline-formula&gt;&lt;tex-math id="M24"&gt;\begin{document}$ p_j\ge q_i+\gamma_i- 1-\frac{2}{N} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, &lt;inline-formula&gt;&lt;tex-math id="M25"&gt;\begin{document}$ i,j = 1,2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; (i.e. the self-diffusion of &lt;inline-formula&gt;&lt;tex-math id="M26"&gt;\begin{document}$ u $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; and &lt;inline-formula&gt;&lt;tex-math id="M27"&gt;\begin{document}$ w $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; are dominant), then the solutions are globally bounded; in particular, different from the results of the single-species chemotaxis system, for the critical case &lt;inline-formula&gt;&lt;tex-math id="M28"&gt;\begin{document}$ p_j = q_i+\gamma_i- 1-\frac{2}{N} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the global boundedness of the solutions can be obtained.&lt;/p&gt;

Radially symmetric stationary solutions for certain chemotaxis systems in higher dimensions: A geometric approach

This paper reports results on the existence, shapes, and asymptotic behavior of positive radially symmetric stationary solutions for several models of chemotaxis in higher dimensions. The systems treated in this paper are the simplest parabolic-elliptic Keller-Segel model and the simplest attraction-repulsion chemotaxis system, in which a positive-valued (resp. negative-valued) solution of one is a negative-valued (resp. positive-valued) solution of the other from symmetry. In particular, the construction of functions satisfying equations that diverge at the endpoints of finite intervals is an interesting result. The key to the discussion is to derive a scalar equation by using a transformation on the averaged mass for the equation satisfied by the radially symmetric stationary solution and to investigate the infinity dynamics as geometric information for the two-dimensional ordinary differential equations derived from it. To achieve this, we use a method that combines classical results from the continuous dynamical systems theory and Poincaré-Lyapunov compactification. In addition, the results for singular solutions are discussed in light of the results of previous studies.

Boundedness criteria for the quasilinear attraction-repulsion chemotaxis system with nonlinear signal production and logistic source

&lt;abstract&gt;&lt;p&gt;This paper deals with the following quasilinear attraction-repulsion chemotaxis system&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{equation*} \left\{ \begin{array}{ll} u_{t} = \nabla\cdot((u+1)^{m}\nabla u-\chi u(u+1)^{\theta-1}\nabla v+\xi u(u+1)^{l-1}\nabla w)+au-bu^{\kappa}, \ &amp;amp;\ \ x\in \Omega, \ t&amp;gt;0, \\ 0 = \Delta v+\alpha u^{\gamma_{1}}-\beta v, \ &amp;amp;\ \ x\in \Omega, \ t&amp;gt;0, \\ 0 = \Delta w+\gamma u^{\gamma_{2}}-\delta w, \ &amp;amp;\ \ x\in \Omega, \ t&amp;gt;0 , \end{array} \right. \end{equation*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;with homogeneous Neumann boundary conditions in a bounded, smooth domain $ \Omega\subset\mathbb{R}^{n} (n\geq 1), $ where $ m, \theta, l \in \mathbb{R} $ and $ \chi, \xi, a, b, \alpha, \beta, \gamma, \delta, \gamma_{1}, \gamma_{2} &amp;gt; 0, \kappa &amp;gt; 1. $ It is proved that if the nonlinear exponents of the system satisfy $ \theta+\gamma_{1} &amp;lt; \max\left\{l+\gamma_{2}, \kappa, m+\frac{2}{n}+1\right\}, $ then the system has globally bounded classical solutions. Furthermore, assume that $ \theta+\gamma_{1} = \max\{l+\gamma_{2}, \kappa\}\geq m+\frac{2}{n}+1, $ if one of the following conditions holds:&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE2"&gt; \begin{document}$ \begin{align*} \mbox{(a)}\ \mbox{when}\ \theta+\gamma_{1} = l+\gamma_{2} = \kappa, &amp;amp; \ \mbox{if}\ \theta\geq l\geq 1 \ \mbox{and} \ \frac{[(\kappa-1-m)n-2](2\alpha \chi-\gamma\xi)}{2(l-1)+(\kappa-1-m)n}&amp;lt;b \\ \ &amp;amp;\mbox{or if}\ \ l \geq \theta\geq 1\ \mbox{and} \ \frac{2\alpha \chi[(\kappa-1-m)n-2]}{2(\theta-1)+(\kappa-1-m)n}&amp;lt;b; \\ \ \mbox{(b)}\ \mbox{when}\ \theta+\gamma_{1} = l+\gamma_{2}&amp;gt;\kappa, &amp;amp; \ \mbox{if}\ \theta\geq l\geq 1 \ \mbox{and} \ 2\alpha \chi\leq \gamma\xi; \\ \ \mbox{(c)}\ \mbox{when}\ \theta+\gamma_{1} = \kappa&amp;gt;l+\gamma_{2}, &amp;amp; \ \mbox{if}\ \theta\geq 1 \ \mbox{and} \ \frac{2\alpha \chi[(\kappa-1-m)n-2]}{2(\theta-1)+(\kappa-1-m)n}&amp;lt;b, \end{align*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;then the classical solutions of the system would be globally bounded. The global boundedness criteria generalize the results established by previous researchers.&lt;/p&gt;&lt;/abstract&gt;

Global boundedness of weak solutions for an attraction-repulsion chemotaxis system with p-Laplacian diffusion and nonlinear production

This paper is concerned with the following attraction-repulsion chemotaxis system with p-Laplacian diffusion and nonlinear production: $ u_{t} = \nabla\cdot(|\nabla u|^{p-2}\nabla u)-\chi \nabla\cdot(u \nabla v)+\xi \nabla\cdot(u \nabla w)+f(u) $, $ v_{t} = \triangle v-\beta v+\alpha u^{k_{1}} $, $ w_{t} = \triangle w-\delta w+\gamma u^{k_{2}} $, under homogenous Neumann boundary condition in a bounded domain $ \Omega \subset \mathbb{R}^{n}(n\geq2) $, with $ \chi, \xi, \alpha,\beta,\gamma,\delta, k_{1}, k_{2} &gt;0, p&gt;1 $. In addition, the function $ f $ satisfying $ f(s)\equiv 0 $ or generalizing the logistic-type source $ f(s) = \kappa s-\mu s^{l} $ for all $ s\geq0 $ with $ \kappa\in \mathbb{R}, \mu&gt;0, l&gt;1 $. It is shown that (ⅰ) When $ f(u)\equiv0 $, if $ p&gt;\frac{n(\max\{k_{1},k_{2}\}+2)}{n+1} $ or $ 1&lt;p\leq\frac{n(\max\{k_{1},k_{2}\}+2)}{n+1} $ with $ \|u_{0}\|_{L^{\frac{(\max\{k_{1},k_{2}\}-p+2)n}{p}}(\Omega)} $ is small, the problem possesses a global bounded weak solution. (ⅱ) When $ f(u) = \kappa u-\mu u^{l} $, if $ \max\{k_{1},k_{2}\}&lt; l-1 $ or $ \max\{k_{1},k_{2}\} = l-1 $ with large $ \mu&gt;0 $, the problem possesses a global bounded weak solution.

Global bounded classical solution for an attraction–repulsion chemotaxis system

In this paper, we deal with the global and boundedness of solutions to a quasilinear attraction–repulsion chemotaxis system with logistic source ut=Δu−χ∇⋅(u(u+1)r−1∇v)+ξ∇⋅(u(u+1)r−1∇w)+f(u),0=Δv−βv+αu,0=Δw−δw+γuin a bounded domain Ω⊂RN(N≥1) with smooth boundary, where χ, ξ, α, β, δ, γ, r are positive constants and f(u)≤a−buη for all u≥0 with some a≥0, b>0 and η≥1. Under the Neumann boundary conditions, if χα=ξγ and η>max{r+(N−2)+N,1} or χα<ξγ and η≥1, we prove that there exists a unique global-in-time and bounded classical solution for all appropriately regular nonnegative initial data, which extends the results of Li and Xiang (2016), Xu and Zheng (2018) and Xie and Zheng (2021).

A quasilinear attraction–repulsion chemotaxis system with logistic source

This paper is concerned with the quasilinear attraction–repulsion chemotaxis system with logistic source ut=∇⋅(D(u)∇u)−∇⋅(Φ(u)∇v)+∇⋅(Ψ(u)∇w)+f(u),(x,t)∈Ω×(0,∞),0=Δv+αu−βv,(x,t)∈Ω×(0,∞),0=Δw+γu−δw,(x,t)∈Ω×(0,∞)under Neumann boundary conditions in a bounded domain Ω⊂RN(N≥1), where D,Φ,Ψ∈C2[0,∞) are nonnegative functions with D(s)≥(s+1)p for s≥0, Φ(s)≤χsq,ξsr≤Ψ(s)≤ζsr for s>1, and the smooth function f satisfies f(s)≤μs(1−sk) for s>0,f(0)≥0. Tian et al. (2016) proved that when q=max{r,k} and q−p≥2N, if one of the following assumptions holds: (i) q=r=k,μ>(αχ−γξ)(1−2N(q−p))/(1+2(q−1)N(q−p)); (ii) q=r>k,αχ−γξ<0; (iii) q=k>r,μ>αχ(1−2N(p−q))/(1+2(q−1)N(p−q)), then the solution of the equations is globally bounded. The present work further shows that the same conclusion still holds for the critical cases: (a) q=r=k,μ=(αχ−γξ)(1−2N(q−p))/(1+2(q−1)N(q−p)); (b) q=r>k,αχ−γξ=0 with q−p<1Nmin{4(N+1)N+2,N+2}; (c) q=k>r,μ=αχ(1−2N(p−q))/(1+2(q−1)N(p−q)).

Global boundedness and asymptotic behavior in an attraction–repulsion chemotaxis system with nonlocal terms

Global boundedness and asymptotic behavior in an attraction–repulsion chemotaxis system with nonlocal terms

Boundedness and finite-time blow-up in a quasilinear parabolic–elliptic–elliptic attraction–repulsion chemotaxis system

This paper deals with the quasilinear attraction-repulsion chemotaxis system \begin{align*} \begin{cases} u_t=\nabla\cdot \big((u+1)^{m-1}\nabla u -\chi u(u+1)^{p-2}\nabla v +\xi u(u+1)^{q-2}\nabla w\big) +f(u), \\[1.05mm] 0=\Delta v+\alpha u-\beta v, \\[1.05mm] 0=\Delta w+\gamma u-\delta w \end{cases} \end{align*} in a bounded domain $\Omega \subset \mathbb{R}^n$ ($n \in \mathbb{N}$) with smooth boundary $\partial\Omega$, where $m, p, q \in \mathbb{R}$, $\chi, \xi, \alpha, \beta, \gamma, \delta>0$ are constants. Moreover, it is supposed that the function $f$ satisfies $f(u)\equiv0$ in the study of boundedness, whereas, when considering blow-up, it is assumed that $m>0$ and $f$ is a function of logistic type such as $f(u)=\lambda u-\mu u^{\kappa}$ with $\lambda \ge 0$, $\mu>0$ and $\kappa>1$ sufficiently close to~$1$, in the radially symmetric setting. In the case that $\xi=0$ and $f(u) \equiv 0$, global existence and boundedness have been proved under the condition $p<m+\frac2n$. Also, in the case that $m=1$, $p=q=2$ and $f$ is a function of logistic type, finite-time blow-up has been established by assuming $\chi\alpha-\xi\gamma>0$. This paper classifies boundedness and blow-up into the cases $p<q$ and $p>q$ without any condition for the sign of $\chi\alpha-\xi\gamma$ and the case $p=q$ with $\chi\alpha-\xi\gamma<0$ or $\chi\alpha-\xi\gamma>0$.

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 ut=∇⋅(D(u)∇u)−∇⋅(G(u)χ(v)∇v)+∇⋅(H(u)ξ(w)∇w),x∈Ω,t>0,vt=d1Δv+αu−βv,x∈Ω,t>0,wt=d2Δw+γu−δw,x∈Ω,t>0,under homogeneous Neumann boundary conditions and initial conditions, where Ω⊂Rn(n≥1) is a bounded domain with smooth boundary, d1,d2,α,β,γ,δ>0 are constants. Also, D,G,H∈C2([0,∞)) fulfill that a0(s+1)m−1≤D(s)≤a1(s+1)m−1 with a0,a1>0 and m∈R; G(0)=0, 0≤G(s)≤b0(s+1)q−1 with b0>0 and q<min{2,m+1}; H(0)=0, 0≤H(s)≤c0(s+1)r−1 with c0>0 and r<min{2,m+1}, and χ,ξ satisfy that 0≤χ(s)≤χ0sk1 with χ0>0 and k1>1; 0≤ξ(s)≤ξ0sk2 with ξ0>0 and k2>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.