On a Two Species Chemotaxis Model with Slow Chemical Diffusion

Negreanu, Mihaela; Tello, J. Ignacio

doi:10.1137/140971853

Cited by 72 publications

(33 citation statements)

References 22 publications

(24 reference statements)

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“…19 Related works which deal with global existence and boundedness in this two-species problem with sensitivity functions can be found in Mizukami 19 and Zhang and Li 23 , and related works which treat the noncompetition case are in previous studies. 18,[20][21][22] These results in the case = 1 are motivated by the results 15,24,25 in the case = 0. Therefore, it seems to be meaningful to study the parabolic-parabolic-elliptic problem reduced by letting = 0.…”

mentioning

confidence: 53%

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

Mizukami

2017

Math Methods in App Sciences

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This paper is concerned with the two-species chemotaxis-competition systemwhere Ω is a bounded domain in R n with smooth boundary Ω, n ≥ 2; i and i are constants satisfying some conditions. The above system was studied in the cases that a 1 , a 2 ∈ (0, 1) and a 1 > 1 > a 2 , and it was proved that global existence and asymptotic stability hold when i i are small. However, the conditions in the above 2 cases strongly depend on a 1 , a 2 , and have not been obtained in the case that a 1 , a 2 ≥ 1. Moreover, convergence rates in the cases that a 1 , a 2 ∈ (0, 1) and a 1 > 1 > a 2 have not been studied. The purpose of this work is to construct conditions which derive global existence of classical bounded solutions for all a 1 , a 2 > 0 which covers the case that a 1 , a 2 ≥ 1, and lead to convergence rates for solutions of the above system in the cases that a 1 , a 2 ∈ (0, 1) and a 1 ≥ 1 > a 2 .

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mentioning

confidence: 53%

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

Mizukami

2017

Math Methods in App Sciences

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“…Applying Gronwall's lemma, we obtain that U þ ¼ U − ¼ 0 (for more details, see for instance the study of Negreanu and Tello 22,23 ). Hence, we have uðtÞ ≤ uðx; tÞ ≤ uðtÞ, for all (x,t) ∈ Ω × (0,∞).…”

Section: Comparison Principle and Asymptotic Behavior Of Solutionsmentioning

confidence: 99%

On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

Negreanu

Tello

Vargas

2019

Math Methods in App Sciences

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We study a parabolic‐elliptic chemotactic PDEs system, which describes the evolution of a biological population “u” and a chemical substance “v” in a bounded domain normalΩ⊂Rn. We consider a growth term of logistic type in the equation of “u” in the form μu(1 − u + f(t,x)). The function “f,” describing the resources of the systems, presents a periodic asymptotic behavior in the sense limt→∞supx∈Ω|f(x,t)−f∗(t)|=0, where f ∗ is independent of x and periodic in time. We study the global existence of solutions and its asymptotic behavior. Under suitable assumptions on the initial data and f ∗, if the constant chemotactic sensitivity χ satisfies χ<μ/2, we obtain that the solution of the system converges to a homogeneous in space and periodic in time function.

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“…with > 0. This function constructed in [11], [12] unifies mathematical structures in the cases > 1 and = 1. This paper is organized as follows.…”

Section: Introductionmentioning

confidence: 96%

“…One of the keys for this strategy is to derive the unified inequality

\begin{matrix} true \\ right \frac{d}{d t} \int_{Ω} u^{p} φ (v) & left + ε p (p - 1) \int_{Ω} u^{p - 2} φ (v) {false| \nabla u false|}^{2} \leq c \int_{Ω} u^{p} φ (v) - r \int_{Ω} u^{p + 1} \frac{φ false(v false)}{{(a + v)}^{k}} \end{matrix}

for some

ε \in [0, 1)

and

c > 0

, where

φ false(s false) : = exp \{- r \int_{η}^{s} \frac{1}{{false(a + τ false)}^{k}} d τ\} false(s \geq η false)

with

r > 0

. This function φ constructed in , unifies mathematical structures in the cases

k > 1

and

k = 1

.…”

Section: Introductionmentioning

confidence: 99%

A unified method for boundedness in fully parabolic chemotaxis systems with signal‐dependent sensitivity

Mizukami

Yokota

2017

Mathematische Nachrichten

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This paper deals with the Keller–Segel system {ut=Δu−∇·false(uχ(v)∇vfalse),x∈Ω,t>0,vt=Δv+u−v,x∈Ω,t>0,where Ω is a bounded domain in Rn with smooth boundary ∂Ω, n≥2; χ is a nonnegative function satisfying χfalse(sfalse)≤K(a+s)−k for some k≥1 and a≥0. In the case that k=1 and a=0, Fujie established global existence of bounded solutions under the condition 0false01, Winkler asserted global existence of bounded solutions for arbitrary K>0. However, there is a gap in the proof. Recently, Fujie tried modifying the proof; nevertheless it also has a gap. It seems to be difficult to show global existence of bounded solutions for arbitrary K>0. Moreover, the condition for K when k>1 cannot connect with the condition when k=1. The purpose of the present paper is to obtain global existence and boundedness under more natural and proper condition for χ and to build a mathematical bridge between the cases k=1 and k>1.

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On a Two Species Chemotaxis Model with Slow Chemical Diffusion

Cited by 72 publications

References 22 publications

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

Boundedness and stabilization in a two‐species chemotaxis‐competition system of parabolic‐parabolic–elliptic type

On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

A unified method for boundedness in fully parabolic chemotaxis systems with signal‐dependent sensitivity

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