Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments

Issa, Tahir Bachar; Shen, Wenxian

doi:10.1016/j.jmaa.2020.124204

Cited by 13 publications

(14 citation statements)

References 37 publications

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“…where Ω ⊂ R N is a bounded smooth domain (see [8,13,16,21,22,24,42,43,44,45], etc.). For example, when a ≡ b ≡ 0 in (1.2), and Ω is a ball in R N with N ≥ 3, it is proved that for any M > 0 there exists positive initial data (u 0 , v 0 ) ∈ C( Ω) × W 1,∞ (Ω) with Ω u 0 = M such that the corresponding solution blows up in finite time (see [43]).…”

Section: Introduction and The Statements Of Main Resultsmentioning

confidence: 99%

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Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$

Shen¹,

Xue²

2021

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The current paper is concerned with the spreading speeds of the following parabolicparabolic chemotaxis model with logistic source on R N ,where χ, a, b, λ, µ are positive constants. Assume b > N µχ 4 . Among others, it is proved that 2 √ a is the spreading speed of the global classical solutions of (0.1) with nonempty compactly supported initial functions, that is,where (u(x, t; u 0 , v 0 ), v(x, t; u 0 , v 0 )) is the unique global classical solution of (0.1) with u(x, 0; u 0 , v 0 ) = u 0 , v(x, 0; u 0 , v 0 ) = v 0 , and supp(u 0 ), supp(v 0 ) are nonempty and compact. It is well known that 2 √ a is the spreading speed of the following Fisher-KPP equation,Hence, if b > N µχ 4 , the chemotaxis neither speeds up nor slows down the spatial spreading in the Fisher-KPP equation.

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Section: Introduction and The Statements Of Main Resultsmentioning

confidence: 99%

“…The particular requirement on the convexity of the bounded domain Ω was later removed in [16] and [45]. Hence finite-time blow-up phenomena in (1.2) can be suppressed to some extent by the logistic source.…”

Section: Introduction and The Statements Of Main Resultsmentioning

confidence: 99%

Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$

Shen¹,

Xue²

2021

Preprint

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“…is a positive constant corresponding to the maximal Sobolev regularity, then (21) admits a unique, smooth and bounded global non-negative solution. Recently, Issa and Shen [10] extended the global existence results obtained in both [34] and [41] to the general full chemotaxis model (21) with f being local as well as nonlocal time and space dependent logistic source. The global existence of classical solutions of ( 21) with τ = 1 and f being logistic source on the whole space R N will be studied somewhere else.…”

Section: Consider the Following General Parabolic-parabolic Chemotaxis Model On Boundedmentioning

confidence: 99%

“…In [30], the authors proved that any such extinction phenomenon must be localized in space, and that the population as a whole always persists, which is called persistence of mass in [30]. Recently, Issa and Shen [10] proved the pointwise persistence phenomena, that is, any globally defined positive solution is bounded below by a positive constant independent of its initial data, which implies that the cell population may become very small at some time and some location, but it persists at any location eventually. For other related works on (3), we refer the readers to [9,14,17,18,21,22,35] and references therein.…”

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Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $

Shen

Xue

2022

DCDS

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In the current paper, we consider the following parabolic-parabolic chemotaxis system with logistic source on <inline-formula><tex-math id="M2">\begin{document}$ \mathbb{R}^{N} $\end{document}</tex-math></inline-formula>,<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation} \begin{cases} u_{t} = \Delta u - \chi\nabla\cdot ( u\nabla v) + u(a-bu),\quad x\in{{\mathbb R}}^N,\\ {v_t} = \Delta v -\lambda v+\mu u,\quad x\in{{\mathbb R}}^N,\,\,\, \end{cases} \;\;\;\;\;\;\;\;\left( 1 \right)\end{equation} $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M3">\begin{document}$ \chi, \ a,\ b,\ \lambda,\ \mu $\end{document}</tex-math></inline-formula> are positive constants and <inline-formula><tex-math id="M4">\begin{document}$ N $\end{document}</tex-math></inline-formula> is a positive integer. We investigate the persistence and convergence in (1). To this end, we first prove, under the assumption <inline-formula><tex-math id="M5">\begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, the global existence of a unique classical solution <inline-formula><tex-math id="M6">\begin{document}$ (u(x,t;u_0, v_0),v(x,t;u_0, v_0)) $\end{document}</tex-math></inline-formula> of (1) with <inline-formula><tex-math id="M7">\begin{document}$ u(x,0;u_0, v_0) = u_0(x) $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M8">\begin{document}$ v(x,0;u_0, v_0) = v_0(x) $\end{document}</tex-math></inline-formula> for every nonnegative, bounded, and uniformly continuous function <inline-formula><tex-math id="M9">\begin{document}$ u_0(x) $\end{document}</tex-math></inline-formula>, and every nonnegative, bounded, uniformly continuous, and differentiable function <inline-formula><tex-math id="M10">\begin{document}$ v_0(x) $\end{document}</tex-math></inline-formula>. Next, under the same assumption <inline-formula><tex-math id="M11">\begin{document}$ b>\frac{N\chi\mu}{4} $\end{document}</tex-math></inline-formula>, we show that persistence phenomena occurs, that is, any globally defined bounded positive classical solution with strictly positive initial function <inline-formula><tex-math id="M12">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula> is bounded below by a positive constant independent of <inline-formula><tex-math id="M13">\begin{document}$ (u_0, v_0) $\end{document}</tex-math></inline-formula> when time is large. Finally, we discuss the asymptotic behavior of the global classical solution with strictly positive initial function <inline-formula><tex-math id="M14">\begin{document}$ u_0 $\end{document}</tex-math></inline-formula>. We show that there is <inline-formula><tex-math id="M15">\begin{document}$ K = K(a,\lambda,N)>\frac{N}{4} $\end{document}</tex-math></inline-formula> such that if <inline-formula><tex-math id="M16">\begin{document}$ b>K \chi\mu $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M17">\begin{document}$ \lambda\geq \frac{a}{2} $\end{document}</tex-math></inline-formula>, then for every strictly positive initial function <inline-formula><tex-math id="M18">\begin{document}$ u_0(\cdot) $\end{document}</tex-math></inline-formula>, it holds that<disp-formula> <label/> <tex-math id="FE2"> \begin{document}$ \lim\limits_{t\to\infty}\big[\|u(x,t;u_0, v_0)-\frac{a}{b}\|_{\infty}+\|v(x,t;u_0, v_0)-\frac{\mu}{\lambda}\frac{a}{b}\|_{\infty}\big] = 0. $\end{document} </tex-math></disp-formula>

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Well-posedness of Keller–Segel systems on compact metric graphs

Shemtaga,

Shen,

Sukhtaiev

2024

J. Evol. Equ.

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Chemotaxis phenomena govern the directed movement of microorganisms in response to chemical stimuli. In this paper, we investigate two Keller–Segel systems of reaction–advection–diffusion equations modeling chemotaxis on thin networks. The distinction between two systems is driven by the rate of diffusion of the chemo-attractant. The intermediate rate of diffusion is modeled by a coupled pair of parabolic equations, while the rapid rate is described by a parabolic equation coupled with an elliptic one. Assuming the polynomial rate of growth of the chemotaxis sensitivity coefficient, we prove local well-posedness of both systems on compact metric graphs, and, in particular, prove existence of unique classical solutions. This is achieved by constructing sufficiently regular mild solutions via analytic semigroup methods and combinatorial description of the heat kernel on metric graphs. The regularity of mild solutions is shown by applying abstract semigroup results to semi-linear parabolic equations on compact graphs. In addition, for logistic-type Keller–Segel systems we prove global well-posedness and, in some special cases, global uniform boundedness of solutions.

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Pointwise persistence in full chemotaxis models with logistic source on bounded heterogeneous environments

Cited by 13 publications

References 37 publications

Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$

Spreading speeds of a parabolic-parabolic chemotaxis model with logistic source on $\mathbb{R}^{N}$

Persistence and convergence in parabolic-parabolic chemotaxis system with logistic source on $ \mathbb{R}^{N} $

Well-posedness of Keller–Segel systems on compact metric graphs

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