Blow-up of nonradial solutions to attraction–repulsion chemotaxis system in two dimensions

Li, Yan; Li, Yuxiang

doi:10.1016/j.nonrwa.2015.12.003

Cited by 54 publications

(43 citation statements)

References 15 publications

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“…It has been proved that when repulsion dominates or cancels attraction in the sense of ξγ ≥ χα, the global classical solution will exists for both τ = 0 [38] and τ = 1 [18,26]. Whereas, if attraction dominates in the sense of ξγ < χα, then the solution of system (1.1) with τ = 0 will blow up in finite time for large initial mass and exist globally with small initial mass in two dimensional spaces [9,23,43]. The large time behavior of solution with small initial data was established in [24].…”

Section: (Communicated By Michael Winkler)mentioning

confidence: 99%

Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

Jin¹,

Xiang²

2018

DCDS-B

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Section: (Communicated By Michael Winkler)mentioning

confidence: 99%

Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

Jin¹,

Xiang²

2018

DCDS-B

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“…Moreover, as for the problem with τ 1 = τ 2 =1 or τ 1 =1, τ 2 =0, some numerical results about global solvability for both cases have been also obtained (see previous studies()). In particular, in four studies,() the global existence of classical solution to was established under the assumptions χ 0 α 1 − ξ 0 α 2 ⩽0 for high dimensions or the assumptions χ 0 α 1 − ξ 0 α 2 >0 and

false| false| u_{0} false| |_{L^{1} false(normalΩ false)} < \frac{4 π}{χ_{0} α_{1} - ξ_{0} α_{2}}

for n =2. Apart from that, if χ 0 α 1 − ξ 0 α 2 >0 and the initial cell mass

\int_{Ω} u_{0}

is large enough, the solution may blow up in n =2.…”

Section: Introductionmentioning

confidence: 99%

Global weak solutions for an attraction‐repulsion system with nonlinear diffusion

Lin

et al. 2017

Math Methods in App Sciences

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This paper deals with the attraction-repulsion chemotaxis system with nonlinear diffusion u t = ∇ · (D(u)∇u) − ∇ · (u (v)∇v) + ∇ · (u (w)∇w), 1 v t = Δv − 1 v + 1 u, 2 w t = Δw − 2 w + 2 u, subject to the homogenous Neumann boundary conditions, in a smooth bounded domain Ω ⊂ R n (n ⩾ 2), where the coefficients i , i , and i ∈ {0, 1}(i = 1, 2) are positive. The function D fulfills D(u) ⩾ C D u m−1 for all u > 0 with certain C D > 0 and m > 1. For the parabolic-elliptic-elliptic case in the sense that 1 = 2 = 0 and = 1, we obtain that for any m > 2 − 2 n and all sufficiently smooth initial data u 0 , the model possesses at least one global weak solution under suitable conditions on the functions and . Under the assumption m > − 2 n , it is also proved that for the parabolic-parabolic-elliptic case in the sense that 1 = 1, 2 = 0, and ⩾ 2, the system possesses at least one global weak solution under different assumptions on the functions and .KEYWORDS attraction-repulsion, boundedness, chemotaxis, logistic source, weak solutionwhich also models the quorum sensing effect in the chemotactic movement (see Painter et al 2 ). Ω ⊂ R n (n ⩾ 2) is a bounded domain with smooth boundary Ω, denotes the differentiation with respect to the outward normal derivative 7368 ;40 7368-7395. : 2 n − for all u ⩾ 0 with some > 0 and c > 0, the corresponding solutions are global and bounded provided that D has algebraic upper and lower growth estimates as u → ∞. Wang et al 47 removed the condition of upper growth estimate of D in Tao et al. 44 More results about Cauchy problem are obtained (see previous studies 8,9,18,25,26,34,[48][49][50] ).Compared to the classical Keller-Segel chemotaxis (1.3), the coupled attraction-repulsion system (1.1) is much less understood because of the lack of Lyapunov functional. As for the system (1.1) with D ≡ 1, ≡ 0 and ≡ 0 , where the coefficients 0 , 0 are positive constants. When 1 , 2 = 0, the authors Hillen et al 42 and Jin et al 51 investigated the 2 −p 1 |Ω| ) C 7 .

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“…For the opposite case θ 1 < 0 (i.e. attraction dominates repulsion), it was shown that the solution of system (5) may blow up in finite time if initial mass is large [5,14] and exist globally for small initial mass [5] in two dimensions. If τ 1 = 1 and τ 2 = 0, Jin and Wang [11] constructed a Lyapunov function…”

mentioning

confidence: 99%

Global stabilization of the full attraction-repulsion Keller-Segel system

Jin

Wang

2020

Discrete &Amp; Continuous Dynamical Systems - A

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Blow-up of nonradial solutions to attraction–repulsion chemotaxis system in two dimensions

Cited by 54 publications

References 15 publications

Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

Repulsion effects on boundedness in a quasilinear attraction-repulsion chemotaxis model in higher dimensions

Global weak solutions for an attraction‐repulsion system with nonlinear diffusion

Global stabilization of the full attraction-repulsion Keller-Segel system

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