Global boundedness and large time behavior of a chemotaxis system with indirect signal absorption

Liu, Yong; Li, Zhongping; Huang, Jinfeng

doi:10.1016/j.jde.2020.05.008

Cited by 27 publications

(11 citation statements)

References 31 publications

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“…for almost every t ∈ (0, T ). Accordingly, the set S : = T > 0 | ∃ strong W 1,q -solution of problem (10) in Ω × (0, T ) that satisfies (36) for a.e. t ∈ (0, T )…”

Section: 1mentioning

confidence: 99%

“…Let N ⊂ (0, T max ) be a set of measure zero, as provided by the definition of S, such that (36) holds for all t ∈ (0, T max )\N , then (36) gives us…”

Section: 1mentioning

confidence: 99%

“…Lv and Wang [38] studied a chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source, and they established the global existence of the solution under some assumptions on the parameters of the logistic source. One can refer to [7,36,42,41,46,47,50,51,75] for further extensions of related chemotaxis models involving ODE-type indirect signal production mechanism and damping source.…”

mentioning

confidence: 99%

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Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

2023

DCDS-B

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This paper deals with the classical solution of the following chemotaxis system with generalized logistic growth and indirect signal production<disp-formula><label/><tex-math id="FE1"> \begin{document}$ \begin{eqnarray} \left\{ \begin{array}{llll} u_t = \epsilon\Delta u-\nabla\cdot(u\nabla v)+ru-\mu u^\theta, &\\ 0 = d_1\Delta v-\beta v+\alpha w, &\\ 0 = d_2\Delta w-\delta w+\gamma u, & \end{array} \right. \end{eqnarray} \quad\quad\quad\quad(1)$ \end{document}</tex-math></disp-formula>and the so-called strong <inline-formula><tex-math id="M1">\begin{document}$ W^{1, q}( \Omega) $\end{document}</tex-math></inline-formula>-solution of hyperbolic-elliptic-elliptic model<disp-formula><label/><tex-math id="FE2"> \begin{document}$ \begin{eqnarray} \left\{ \begin{array}{llll} u_t = -\nabla\cdot(u\nabla v)+ru-\mu u^\theta, &\\ 0 = d_1\Delta v-\beta v+\alpha w, &\\ 0 = d_2\Delta w-\delta w+\gamma u, & \end{array} \right. \end{eqnarray} \quad\quad\quad\quad(2)$ \end{document}</tex-math></disp-formula>in arbitrary bounded domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^n $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ n\geq1 $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M4">\begin{document}$ r, \mu, d_1, d_2, \alpha, \beta, \gamma, \delta>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ \theta>1 $\end{document}</tex-math></inline-formula>. Via applying the viscosity vanishing method, we first prove that the classical solution of (1) will converge to the strong <inline-formula><tex-math id="M6">\begin{document}$ W^{1, q}( \Omega) $\end{document}</tex-math></inline-formula>-solution of (2) as <inline-formula><tex-math id="M7">\begin{document}$ \epsilon\rightarrow0 $\end{document}</tex-math></inline-formula>. After structuring the local well-pose of (2), we find that the strong <inline-formula><tex-math id="M8">\begin{document}$ W^{1, q}( \Omega) $\end{document}</tex-math></inline-formula>-solution will blow up in finite time with non-radial symmetry setting if <inline-formula><tex-math id="M9">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula> is a bounded convex domain, <inline-formula><tex-math id="M10">\begin{document}$ \theta\in(1, 2] $\end{document}</tex-math></inline-formula>, and the initial data is suitable large. Moreover, for any positive constant <inline-formula><tex-math id="M11">\begin{document}$ M $\end{document}</tex-math></inline-formula> and the classical solution of (1), if we add another hypothesis that there exists positive constant <inline-formula><tex-math id="M12">\begin{document}$ \epsilon_0(M) $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M13">\begin{document}$ \epsilon\in(0,\ \epsilon_0(M)) $\end{document}</tex-math></inline-formula>, then the classical solution of (1) can exceed arbitrarily large finite value in the sense: one can find some points <inline-formula><tex-math id="M14">\begin{document}$ \left(\tilde{x}, \tilde{t}\right) $\end{document}</tex-math></inline-formula> such that <inline-formula><tex-math id="M15">\begin{document}$ u(\tilde{x}, \tilde{t})>M $\end{document}</tex-math></inline-formula>.

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“…for almost every t ∈ (0, T ). Accordingly, the set S : = T > 0 | ∃ strong W 1,q -solution of problem (10) in Ω × (0, T ) that satisfies (36) for a.e. t ∈ (0, T )…”

Section: 1mentioning

confidence: 99%

“…Let N ⊂ (0, T max ) be a set of measure zero, as provided by the definition of S, such that (36) holds for all t ∈ (0, T max )\N , then (36) gives us…”

Section: 1mentioning

confidence: 99%

mentioning

confidence: 99%

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Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

2023

DCDS-B

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“…Lv and Wang [26] studied a chemotaxis system with signal-dependent motility, indirect signal production and generalized logistic source, and they established the global existence of the solution under some assumptions on the parameters of the logistic source. One can refer to [6,25,29,30,34,39,40,64] for further extensions of related chemotaxis models involving ODE-type indirect signal production mechanism and damping source.…”

Section: Introductionmentioning

confidence: 99%

Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

2020

Preprint

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This paper deals with the classical solution of following chemotaxis system with indirect signal production, where χ, r, µ, d 1 , α, β, γ, λ > 0 and θ > 1, we find:(i) for ǫ ∈ (0, 1), d 2 > 0 and τ 1 = τ 2 = 0, if the initial data u 0 L θ (Ω) is sufficiently large, then for the associated solution of system and any positive constant M , one can find some points x, t such that u(x, t) > M.(0.1) (ii) for ǫ, τ 1 , τ 2 > 0, d 2 ≥ 0 and positive constant K, if F(u 0 , v 0 ) ≤ −K, here F is the Lyapunov functional to the system without damping source ru − µu θ , then (0.1

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“…, the solution of Equation (1.6) admits a globally bounded classical solution, which converges to the constant steady state ( ū0 , 0, ū0 𝛿 ) as time goes to infinity. As 𝑓(𝑢) = 𝜇(𝑢 − 𝑢 𝑟 ) with 𝜇 ≥ 0 and 𝑟 > 1 for all 𝑢 ≥ 0, Liu et al [26] show that if 𝑟 > 𝑁 2…”

mentioning

confidence: 99%

Global boundedness and asymptotic behavior in a chemotaxis system with signal‐dependent motility and indirect signal absorption

Ren

Shi

2022

Z Angew Math Mech

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In this paper we investigate the following chemotaxis system with signal‐dependent motility and indirect signal absorption leftutgoodbreak=normalΔ(γfalse(vfalse)u)goodbreak+μu(1−u),leftx∈normalΩ,tgoodbreak>0,leftvtgoodbreak=normalΔvgoodbreak−vw,leftx∈normalΩ,tgoodbreak>0,leftwtgoodbreak=−δwgoodbreak+u,leftx∈normalΩ,t>0$$\begin{equation} {\left\lbrace \def\eqcellsep{&}\begin{array}{lll}u_t=\Delta (\gamma (v)u)+\mu u(1-u), &x\in \Omega , t>0, \\[3pt] v_t=\Delta v-vw, & x\in \Omega , t>0, \\[3pt] w_t=-\delta w+u, & x\in \Omega , t>0 \end{array} \right.} \end{equation}$$in a bounded domain with smooth boundary. We present the global existence of classical solutions to the model (*) in two space dimensions with logistic source and in any dimension for small initial data without logistic source. In addition, the asymptotic behavior of the solutions is studied.

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Global boundedness and large time behavior of a chemotaxis system with indirect signal absorption

Cited by 27 publications

References 31 publications

Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

Emergence of lager densities in chemotaxis system with indirect signal production and non-radial symmetry case

Global boundedness and asymptotic behavior in a chemotaxis system with signal‐dependent motility and indirect signal absorption

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