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

Chiyo, Yutaro; Yokota, Tomomi

doi:10.1007/s00033-022-01695-y

Cited by 19 publications

(14 citation statements)

References 47 publications

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“…On the other hand, as to the quasilinear version, such as (1.1), of the above system (1.4) with τ = 0, there are several studies. Indeed, boundedness and blow-up were classified by the size of p, q in [4] and stabilization was obtained in [1,3].…”

Section: Y Chiyomentioning

confidence: 99%

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

Chiyo¹

2023

Arch. Math. (Brno)

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Section: Y Chiyomentioning

confidence: 99%

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

Chiyo¹

2023

Arch. Math. (Brno)

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“…In [33], when f (u) = λu − µu γ+1 , λ ≥ 0, µ > 0, some global "very weak" solutions of (1) were constructed for rather arbitrary initial data under the assumption that γ > 1 − 1 n and n ≥ 2. Winkler [34] proved that the solutions blow up in finite time if 1 ≤ γ < 3 2 + 1 2n−2 , n ≥ 5. In [38], Winkler also obtained the finite-time blow-up of solution under the conditions 1 < γ < 7 6 , n = 3, 4, or 1 < γ < 1 + 1 2n−2 , n ≥ 5.…”

Section: Runlin Hu Pan Zheng and Zhangqin Gaomentioning

confidence: 99%

“…During the past few decades, for the attraction-repulsion chemotaxis system, many mathematicians have obtained lots of results about boundedness, finite-time blow-up and asymptotic behavior of solutions in [3,32,23,41]. Particularly, these results in [15,16,20,30] indicate that the global existence and boundedness of solutions are derived due to the inhibition of repulsion to the attraction.…”

Section: Runlin Hu Pan Zheng and Zhangqin Gaomentioning

confidence: 99%

Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production

Zheng

Gao

2022

EECT

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This paper deals with a quasilinear parabolic-elliptic chemo-repulsion system with nonlinear signal production<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{eqnarray*} \label{1a} \left\{ \begin{split}{} & u_t = \nabla\cdot(\phi(u)\nabla u)+\chi\nabla\cdot(u(u+1)^{\alpha-1}\nabla v)+f(u), & (x,t)\in \Omega\times (0,\infty), \\ & 0 = \Delta v-v+u^{\beta}, & (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}(n\geq1), $\end{document}</tex-math></inline-formula> where <inline-formula><tex-math id="M2">\begin{document}$ \chi,\beta>0,\alpha\in\mathbb{R}, $\end{document}</tex-math></inline-formula> the nonlinear diffusion <inline-formula><tex-math id="M3">\begin{document}$ \phi\in C^{2}([0,\infty)) $\end{document}</tex-math></inline-formula> satisfies <inline-formula><tex-math id="M4">\begin{document}$ \phi(u)\geq(u+1)^{m} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M5">\begin{document}$ m\in\mathbb{R}, $\end{document}</tex-math></inline-formula> and the function <inline-formula><tex-math id="M6">\begin{document}$ f\in C^{1}([0,\infty)) $\end{document}</tex-math></inline-formula> is a generalized growth term.<inline-formula><tex-math id="M7">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> When <inline-formula><tex-math id="M8">\begin{document}$ f\equiv0, $\end{document}</tex-math></inline-formula> it is shown that the solution of the above system is global and uniformly bounded for all <inline-formula><tex-math id="M9">\begin{document}$ \chi,\beta>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ m,\alpha\in\mathbb{R} $\end{document}</tex-math></inline-formula>.<inline-formula><tex-math id="M11">\begin{document}$ \bullet $\end{document}</tex-math></inline-formula> When <inline-formula><tex-math id="M12">\begin{document}$ f\not\equiv0 $\end{document}</tex-math></inline-formula> and assume that <inline-formula><tex-math id="M13">\begin{document}$ f(u)\leq ku-bu^{\gamma+1} $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M14">\begin{document}$ k,b,\gamma>0, $\end{document}</tex-math></inline-formula> it is proved that the solution of the above system is also global and uniformly bounded for all <inline-formula><tex-math id="M15">\begin{document}$ \chi,\beta>0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M16">\begin{document}$ m,\alpha\in\mathbb{R}. $\end{document}</tex-math></inline-formula>

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“…In this section we assume that p, q, χ, ξ, α, γ fulfill either (1.3) or (1.4), and that u 0 satisfies (1.2). Then we denote by (u, v, w) the global classical solution of the problem (1.1) given in [5].…”

Section: Stabilizationmentioning

confidence: 99%

“…where m, p, q ∈ R, due to the quasilinear structure of nonlinearities, it is no longer valid to use the transformation z := χv − ξw. Meanwhile, global existence and boundedness of solutions have already been shown in the parabolic-elliptic-elliptic case under the condition that p < q, or p = q and χα − ξγ < 0 ( [5]). However, the following question remains:…”

Section: Introductionmentioning

confidence: 99%

Stabilization for small mass in a quasilinear parabolic--elliptic--elliptic attraction-repulsion chemotaxis system with density-dependent sensitivity: repulsion-dominant case

Chiyo¹

2022

Preprint

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This paper deals with the quasilinear attraction-repulsion chemotaxis systemwith smooth boundary ∂Ω, where m, p, q ∈ R, χ, ξ, α, β, γ, δ > 0 are constants. In the case that m = 1 and p = q = 2, when χα − ξγ < 0 and β = δ, Tao-Wang (Math. Models Methods Appl. Sci.; 2013; 23; 1-36) proved that global bounded classical solutions toward the spatially constant equilibrium (u 0 , α β u 0 , γ δ u 0 ) via the reduction to the Keller-Segel system by using the transformation z := χv − ξw, where u 0 is the spatial average of the initial data u 0 . However, since the above system involves nonlinearities, the method is no longer valid. The purpose of this paper is to establish that global bounded classical solutions converge to the spatially constant equilibrium (u 0 , α β u 0 , γ δ u 0 ).

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

Cited by 19 publications

References 47 publications

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

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

Boundedness of solutions in a quasilinear chemo-repulsion system with nonlinear signal production

Stabilization for small mass in a quasilinear parabolic--elliptic--elliptic attraction-repulsion chemotaxis system with density-dependent sensitivity: repulsion-dominant case

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