A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

Wang, Liangchen; Mu, Chunlai

doi:10.3934/dcdsb.2020114

Cited by 21 publications

(18 citation statements)

References 54 publications

(87 reference statements)

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“…In case τ = 0, the global solution was obtained in [37] if µ i (i = 1, 2) were sufficiently large; the stabilization was also obtained if µ i are sufficiently large; the results were further investigated by Wang and Mu [39].…”

Section: (I) Single-species Chemotaxis Modelmentioning

confidence: 97%

On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

Pan¹,

Wang²

2022

DCDS-B

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This paper deals with the following two-species chemotaxis system with nonlinear diffusion, sensitivity, signal secretion and (without or with) logistic source        ut = ∇ • (D 1 (u)∇u − S 1 (u)∇v) + f 1 (u), x ∈ Ω, t > 0, vt = ∆v − v + g 1 (w), x ∈ Ω, t > 0, wt = ∇ • (D 2 (w)∇w − S 2 (w)∇z) + f 2 (w), x ∈ Ω, t > 0, zt = ∆z − z + g 2 (u), x ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n with n ≥ 2. The diffusion functions D i (s) ∈ C 2 ([0, ∞)) and the chemotactic sensitivity functions S i (s) ∈ C 2 ([0, ∞)) are given by

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Section: (I) Single-species Chemotaxis Modelmentioning

confidence: 97%

On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

Pan¹,

Wang²

2022

DCDS-B

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“…Recently, Wang et al 29 proved that in n ≤ 3, global boundedness solution of () was obtained when μ i ( i =1,2) were large enough, and the large time behavior was obtained under the condition that μ 1 and μ 2 were sufficiently large. Moreover, global boundedness of solution in n =3 was further investigated in Pan et al 30 In addition, some other results about the boundedness and large time behavior of two‐species chemotaxis system were obtained in previous studies 31‐40 …”

Section: Introductionmentioning

confidence: 96%

“…Moreover, global boundedness of solution in n = 3 was further investigated in Pan et al 30 In addition, some other results about the boundedness and large time behavior of two-species chemotaxis system were obtained in previous studies. [31][32][33][34][35][36][37][38][39][40] In this paper, we deal with the following system, which describes the communication between macrophages and breast tumor cells through a near chemical signalling loop:…”

Section: Introductionmentioning

confidence: 99%

Boundedness in a three‐dimensional two‐species and two‐stimuli chemotaxis system with chemical signalling loop

Pan

Wang

Zhang

2020

Math Methods in App Sciences

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The following two‐species chemotaxis system with a chemical signalling loop under Lotka–Volterra competitive kinetics ut=normalΔu−χ1∇·false(u∇vfalse)+μ1ufalse(1−u−a1wfalse),x∈normalΩ,.3emt>0,vt=normalΔv−v+w,x∈normalΩ,.3emt>0,wt=normalΔw−χ2∇·false(w∇zfalse)−χ3∇·false(w∇vfalse)+μ2wfalse(1−w−a2ufalse),x∈normalΩ,.3emt>0,zt=normalΔz−z+u,x∈normalΩ,.3emt>0, is considered in a bounded domain normalΩ⊂ℝ3 with homogeneous Neumann boundary conditions, and the parameters μi,ai>0 (i=1,2) and χj>0(j=1,2,3), the initial data are nonnegative and satisfy false(u0,v0,w0,z0false)∈C0false(truenormalΩ¯false)×C1false(truenormalΩ¯false)×C0false(truenormalΩ¯false)×C1false(truenormalΩ¯false). Global boundedness solution of this system is proved under the conditions μ1>max{}()10+2()1+34χ12+χ22+χ32+1,0.3em.3em2χ12+()4+3()10+22χ12+χ22+χ32χ12 and μ2>max{2χ32+()4+3()10+22χ12+χ22+χ32χ32+()10+2()1+34χ12+χ22+χ32+1,}2χ22+()4+3()10+22χ12+χ22+χ32χ22.

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“…Choosing h 1 (u 2 ) = u 2 , h 2 (u 1 ) = u 1 , χ 11 , χ 22 are two constants, when µ 1 = µ 2 = 0, the global boundedness and blow-up of solutions have been considered in [6,11,20]. When µ 1 , µ 2 = 0, for the fully parabolic case, the global boundedness and large time behavior for n ≤ 2 and n = 3 were detected in [3] and [8] respectively; as for the parabolic-elliptic case, for all n ≥ 1, the global boundedness and asymptotic behavior were obtained in [21,12]; afterwards, the results in [21,12] were partially improved by Wang et al in [18].…”

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confidence: 99%

“…To handle the first term on the right side of (66), we notice that n = 2, accordingly, in view of (18), the following inequality L 2 (Ω) + C 13 for all t ∈ (0, T max ) and > 0 (68) with C 13 > 0. Therefore, utilizing the Hölder inequality, the Gagliardo-Nirenberg inequality and the Young inequality, it follows from ( 12), (67) as well as ( 68) that…”

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confidence: 99%

Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

Huang

2021

era

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In this work, the fully parabolic chemotaxis-competition system with loop<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \left\{ \begin{array}{llll} &\partial_{t} u_{1} = d_1\Delta u_{1}-\nabla\cdot(u_{1}\chi_{11}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{1}\chi_{12}(v_{2})\nabla v_{2}) +\mu_{1}u_{1}(1-u_{1}-a_{1}u_{2}),\\ &\partial_{t} u_{2} = d_2\Delta u_{2}-\nabla\cdot(u_{2}\chi_{21}(v_{1})\nabla v_{1}) \\& \qquad -\nabla\cdot(u_{2}\chi_{22}(v_{2})\nabla v_{2}) +\mu_{2}u_{2}(1-u_{2}-a_{2}u_{1}), \\ &\partial_t v_1 = d_3\Delta v_{1}-\lambda_{1} v_{1}+h_1(u_{1}, u_{2}), \\ &\partial_t v_2 = d_4\Delta v_{2}-\lambda_{2} v_{2}+h_2(u_{1}, u_{2}) \\ \end{array} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula>is considered under the homogeneous Neumann boundary condition, where <inline-formula><tex-math id="M1">\begin{document}$ x\in\Omega, t>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset \mathbb{R}^{n} (n\leq 3) $\end{document}</tex-math></inline-formula> is a bounded domain with smooth boundary. For any regular nonnegative initial data, it is proved that if the parameters <inline-formula><tex-math id="M3">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula> are sufficiently large, then the system possesses a unique and global classical solution for <inline-formula><tex-math id="M4">\begin{document}$ n\leq 3 $\end{document}</tex-math></inline-formula>. Specifically, when <inline-formula><tex-math id="M5">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, the global boundedness can be attained without any constraints on <inline-formula><tex-math id="M6">\begin{document}$ \mu_1, \mu_2 $\end{document}</tex-math></inline-formula>.

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A new result for boundedness and stabilization in a two-species chemotaxis system with two chemicals

Abstract: This paper deals with the following competitive two-species chemotaxis system with two chemicals     1. Introduction. In this paper, we consider the two-species chemotaxiscompetition system with two chemicals

Cited by 21 publications

References 54 publications

On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

On a quasilinear fully parabolic two-species chemotaxis system with two chemicals

Boundedness in a three‐dimensional two‐species and two‐stimuli chemotaxis system with chemical signalling loop

Global boundedness for a chemotaxis-competition system with signal dependent sensitivity and loop

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