Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant

Baghaei, Khadijeh; Khelghati, Ali

doi:10.1016/j.crma.2017.04.009

Cited by 27 publications

(10 citation statements)

References 11 publications

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“…Remark Note that the result obtained in Theorem improves the result obtained in [] with

f (x, u, v) = u v

under the condition

0 < false∥ v_{0} ∥_{L^{\infty} false(normalΩ false)} < \frac{1}{6 χ (n + 1)}

and coincides with our result in []. More precisely, we can write

{trueprefixlim}_{n \to \infty} \frac{\frac{π}{χ} \sqrt{\frac{2}{n}}}{\frac{1}{6 χ false(n + 1 false)}} = + \infty

.…”

Section: The Proof Of Our Main Resultssupporting

confidence: 89%

“…More precisely, by comparing the two conditions, we can write

{trueprefixlim}_{n \to \infty} \frac{\frac{π}{χ} \sqrt{\frac{2}{n}}}{\frac{1}{6 χ false(n + 1 false)}} = + \infty

. This result also coincides with our results in []. In the next section, we prove our main results.…”

Section: Introductionsupporting

confidence: 87%

“…Also, they proved that the solution to this problem satisfies the following convergence properties:

({, \begin{matrix} 0true u (., t) \overset{}{\to} {\bar{u}}_{0} : = \frac{1}{false| normalΩ false|} \int_{Ω} u_{0} (x) & in L^{\infty} false(normalΩ false) as t \overset{}{\to} \infty, \\ v false(., t false) \overset{}{\to} 0 & in L^{\infty} false(normalΩ false) as t \overset{}{\to} \infty . \end{matrix})

Besides, they proved the existence of bounded weak solutions in the three‐dimensional case provided that the initial data be arbitrarily large . Our recent results to this problem show that the classical solutions are global and bounded provided that

0 < false∥ v_{0} ∥_{L^{\infty} false(normalΩ false)} < \frac{π}{χ \sqrt{2 (n + 1)}}

. The same result holds in the presence of logistic source

g false(s false) = a s - b s^{2}

.…”

Section: Introductionmentioning

confidence: 96%

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Boundedness of classical solutions for a chemotaxis model with rotational flux terms

Khelghati

Baghaei

2018

Z Angew Math Mech

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In this paper, we study the following chemotaxis system with rotational flux terms:where 0 is some non-decreasing function.We prove that the classical solutions to the above system are uniformly in-timebounded if there exists a smooth function ( ) with ′ ( ) ≥ 0 such that for some > 2 the matrix-valued function:be a negative semi-definite matrix. Here, denotes the transpose of and I is an × identity matrix. We show that the preceding matrix-valued function is a negative semi-definite matrix provided thatThese results extend the recent results obtained by Li et al. (Math. Models Methods Appl. Sci.) (2015) and Zhang (Math. Nachr.) (2016).We also study the special case = I with > 0. The above matrix in this case is written as:[ 1 4 ( − 1) 2 + 1 ( − 1) ( ′ ( )) 2 − 1 ′′ ( ) ] I ∶= ( )I . 1864

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“…Remark Note that the result obtained in Theorem improves the result obtained in [] with

f (x, u, v) = u v

under the condition

0 < false∥ v_{0} ∥_{L^{\infty} false(normalΩ false)} < \frac{1}{6 χ (n + 1)}

and coincides with our result in []. More precisely, we can write

{trueprefixlim}_{n \to \infty} \frac{\frac{π}{χ} \sqrt{\frac{2}{n}}}{\frac{1}{6 χ false(n + 1 false)}} = + \infty

.…”

Section: The Proof Of Our Main Resultssupporting

confidence: 89%

“…More precisely, by comparing the two conditions, we can write

{trueprefixlim}_{n \to \infty} \frac{\frac{π}{χ} \sqrt{\frac{2}{n}}}{\frac{1}{6 χ false(n + 1 false)}} = + \infty

. This result also coincides with our results in []. In the next section, we prove our main results.…”

Section: Introductionsupporting

confidence: 87%

“…Also, they proved that the solution to this problem satisfies the following convergence properties:

({, \begin{matrix} 0true u (., t) \overset{}{\to} {\bar{u}}_{0} : = \frac{1}{false| normalΩ false|} \int_{Ω} u_{0} (x) & in L^{\infty} false(normalΩ false) as t \overset{}{\to} \infty, \\ v false(., t false) \overset{}{\to} 0 & in L^{\infty} false(normalΩ false) as t \overset{}{\to} \infty . \end{matrix})

0 < false∥ v_{0} ∥_{L^{\infty} false(normalΩ false)} < \frac{π}{χ \sqrt{2 (n + 1)}}

. The same result holds in the presence of logistic source

g false(s false) = a s - b s^{2}

.…”

Section: Introductionmentioning

confidence: 96%

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Boundedness of classical solutions for a chemotaxis model with rotational flux terms

Khelghati

Baghaei

2018

Z Angew Math Mech

Self Cite

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“…When 𝑁 = 3, Tao and Winkler [48] studied the system (1.3) in bounded convex domains and they proved that for arbitrarily large initial data, the problem admits at least one global weak solution for which there exists 𝑇 > 0 such that (𝑢, 𝑣) is bounded and smooth in Ω × (𝑇, ∞) and such solution converges to the constant steady state ( ū0 , 0). Baghaei and Khelghati [1] improved the result in Refs. [48,62] and showed that if ‖𝑣 0 ‖ 𝐿 ∞ (Ω) ≤ 𝜋 𝜒 √ 2(𝑁+1)…”

supporting

confidence: 57%

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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“…Tao [19] proved that whenever the corresponding initial data are sufficiently smooth and satisfy v 0 L ∞ (Ω) ≤ 1 6(n+1) , then there exists a global classical solution of (1.2). In [1] this condition has then been improved; it is sufficient to require v 0 L ∞ (Ω) < π √ 2(n+1) .…”

Section: Introductionmentioning

confidence: 99%

Analysis of a chemotaxis model with indirect signal absorption

Fuest

2019

Journal of Differential Equations

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We consider the chemotaxis modelIf either n ≤ 2 or v0 L ∞ (Ω) ≤ 1 3n we show the existence of a unique global classical solution (u, v, w) and convergence of (u(·, t), v(·, t), w(·, t)) towards a spatially constant equilibrium, as t → ∞.The proof of global existence for the case n ≤ 2 relies on a bootstrap procedure. As a starting point we derive a functional inequality for a functional being sublinear in u, which appears to be novel in this context. behavior AMS Classification (2010): 35K55 (primary), 35A01, 35K40, 92C17 (secondary) * fuestm@math.uni-paderborn.de Ω ∇u · ∇v − Ω ∇v · ∇w instead. Even ignoring the fact that w might not be smooth enough to justify the calculation, it is not clear at all how to handle these terms.

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Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant

Abstract: Partial differential equations Boundedness of classical solutions for a chemotaxis model with consumption of chemoattractant Les solutions classiques d'un modèle de chimiotaxie avec consommation de chimioattracteurs sont bornées

Cited by 27 publications

References 11 publications

Boundedness of classical solutions for a chemotaxis model with rotational flux terms

Boundedness of classical solutions for a chemotaxis model with rotational flux terms

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

Analysis of a chemotaxis model with indirect signal absorption

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