Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production

Ding, Mengyao; Wang, Wei

doi:10.3934/dcdsb.2018328

Cited by 36 publications

(14 citation statements)

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“…where a 0 , b 0 > 0, α, β ∈ R, Ding and Wang [4] studied the global boundedness of solutions in (1.6) if α + β < 3 for n = 1 or α + β < 4 n for n ≥ 2 when µ = 0, τ = 1 and h(v, w) = −v + w. By comparing with the previous results in [16], it is not difficult to see that the indirect signal production could be helpful for the global boundedness of solutions. When h(v, w) = −vw, µ, τ = 0 in (1.6) and D, S satisfy (1.2), Fuest [6] proved that the solution (u, v, w) of system (1.6) is globally bounded and converges to a constant equilibrium if either n ≤ 2 or ||v 0 || L ∞ (Ω) ≤ 1 3n .…”

Section: S(s) D(s)mentioning

confidence: 99%

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On an exponentially decaying diffusive chemotaxis system with indirect signals

Zheng

Xing

2022

CPAA

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This paper deals with an exponentially decaying diffusive chemotaxis system with indirect signal production or consumption<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} &u_t = \nabla\cdot(D(u)\nabla u)-\nabla\cdot(S(u)\nabla v), &(x,t)\in \Omega\times (0,\infty), \\ &v_t = \Delta v+h(v,w), &(x,t)\in \Omega\times (0,\infty), \\ &w_t = \Delta w- w+u, &(x,t)\in \Omega\times (0,\infty), \end{split} \right. \end{eqnarray*} $\end{document} </tex-math></disp-formula>under homogeneous Neumann boundary conditions in a smoothly bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset \mathbb{R}^{n} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M2">\begin{document}$ n\geq2 $\end{document}</tex-math></inline-formula>, where the nonlinear diffusivity <inline-formula><tex-math id="M3">\begin{document}$ D $\end{document}</tex-math></inline-formula> and chemosensitivity <inline-formula><tex-math id="M4">\begin{document}$ S $\end{document}</tex-math></inline-formula> are supposed to satisfy<disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ K_{1}e^{-\beta^{-}s}\leq D(s) \leq K_{2}e^{-\beta^{+}s} \;\;\;{\rm{and}}\;\;\;\frac{D(s)}{S(s)}\geq K_{3}s^{-\alpha}+\gamma, $\end{document} </tex-math></disp-formula>with the constants <inline-formula><tex-math id="M5">\begin{document}$ \beta^{-}\geq \beta^{+}>0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ K_{1},K_{2},K_{3}>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ \alpha,\gamma\geq0 $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M8">\begin{document}$ h(v,w) = -v+w $\end{document}</tex-math></inline-formula>, we study the global existence and boundedness of solutions for the above system provided that <inline-formula><tex-math id="M9">\begin{document}$ \alpha\in[0,\frac{2}{n}) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M10">\begin{document}$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M11">\begin{document}$ \gamma>1 $\end{document}</tex-math></inline-formula> and the initial mass of <inline-formula><tex-math id="M12">\begin{document}$ u_{0} $\end{document}</tex-math></inline-formula> is small enough. Moreover, it is proved that the global bounded solution <inline-formula><tex-math id="M13">\begin{document}$ (u,v,w) $\end{document}</tex-math></inline-formula> converges to <inline-formula><tex-math id="M14">\begin{document}$ (\overline{u_{0}},\overline{u_{0}},\overline{u_{0}}) $\end{document}</tex-math></inline-formula> in the <inline-formula><tex-math id="M15">\begin{document}$ L^{\infty} $\end{document}</tex-math></inline-formula>-norm as <inline-formula><tex-math id="M16">\begin{document}$ t\rightarrow \infty $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M17">\begin{document}$ \overline{u_{0}} = \frac{1}{|\Omega|}\int_{\Omega}u_{0}(x)dx $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M18">\begin{document}$ h(v,w) = -vw $\end{document}</tex-math></inline-formula>, it is shown that this system possesses a unique uniformly bounded classical solution if <inline-formula><tex-math id="M19">\begin{document}$ \alpha\geq0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M20">\begin{document}$ \gamma>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M21">\begin{document}$ \beta^{-}\geq \beta^{+}>\frac{n}{2} $\end{document}</tex-math></inline-formula>. Furthermore, if <inline-formula><tex-math id="M22">\begin{document}$ n = 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M23">\begin{document}$ \alpha\geq0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M24">\begin{document}$ \gamma\geq0 $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M25">\begin{document}$ \beta^{-}\geq \beta^{+}>\varepsilon $\end{document}</tex-math></inline-formula> with some <inline-formula><tex-math id="M26">\begin{document}$ \varepsilon>0 $\end{document}</tex-math></inline-formula>, we only obtain the global existence of solutions for the above system.

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Section: S(s) D(s)mentioning

confidence: 99%

“…Proof. The proof of local existence follows from the Banach fixed point theorem and standard parabolic regularity theory (see [3,4,9]), we do not explain more details here. And integrating the first equation of (1.8) over x ∈ Ω, we can obtain (2.2).…”

Section: ω|mentioning

confidence: 99%

On an exponentially decaying diffusive chemotaxis system with indirect signals

Zheng

Xing

2022

CPAA

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“…Then the work in [18] improves the result of [14] to a general case when n ≥ 2 and α ≥ n 2 . Some similar models can refer to [5,6,28]. In this paper, we consider the three-component parabolic…”

Section: Introductionmentioning

confidence: 99%

“…Besides, large time behavior of solutions to the system in [45] has been also studied with α > n+2 4 and n ≥ 2. Inspired by the above works [5,14,18,28,33,45], we consider the quasilinear parabolic chemotaxis system. Our main result on the global existence and boundedness of solutions to system (1.5) is as follows: Theorem 1.1 Let be a bounded domain in R n (n ≥ 3) with smooth boundary ∂ .…”

Section: Introductionmentioning

confidence: 99%

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

2021

Acta Appl Math

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“…Later, Hu and Tao [13] studied the boundedness and large time behavior for a parabolic-parabolic ODE chemotaxis-growth system with indirect signal production. Moreover, Ding and Wang [7] investigated global boundedness in the quasilinear fully parabolic chemotaxis model with indirect signal production. Wang [30] studied the boundedness for the quasilinear fully parabolic chemotaxis-growth system with indirect signal production, and applied the results into the quasilinear attraction-repulsion chemotaxis model with logistic source.…”

mentioning

confidence: 99%

Asymptotic stability in a chemotaxis-competition system with indirect signal production

Zheng¹

2021

Discrete &Amp; Continuous Dynamical Systems - A

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Global boundedness in a quasilinear fully parabolic chemotaxis system with indirect signal production

Cited by 36 publications

References 0 publications

On an exponentially decaying diffusive chemotaxis system with indirect signals

On an exponentially decaying diffusive chemotaxis system with indirect signals

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

Asymptotic stability in a chemotaxis-competition system with indirect signal production

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