Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant

Negreanu, Mihaela; Tello, J. Ignacio

doi:10.1016/j.jde.2014.11.009

Cited by 89 publications

(47 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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: 55%

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

Mizukami

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

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 .

show abstract

mentioning

confidence: 55%

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

Mizukami

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…x ∈ Ω, t > 0, (1.4) which has been extensively investigated by many authors. For instance, the global existence and asymptotic behavior has been constructed to (1.4) with a 1 = a 2 = 0 [24,25] when 0 ≤ d 3 < 1 and this restriction was removed by Mizukami and Yokota [18]. If n ≤ 2, Bai and Winkler [1] showed (1.4) has global bounded solution and the n-dimensional setting [14,15,48]; moreover, for any global bounded solution, the asymptotic behavior was constructed if µ 1 and µ 2 were sufficiently large, this conditions were improved by Mizukami [19].…”

Section: 2)mentioning

confidence: 99%

Boundedness and stabilization in a two-species chemotaxis system with two chemicals

Wang¹,

Zhang²,

Mu³

et al. 2020

Discrete &Amp; Continuous Dynamical Systems - B

View full text Add to dashboard Cite

This paper deals with the two-species chemotaxis system with two chemicalsx ∈ Ω, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n (n ≥ 1), where the parameters d 1 , d 2 , d 3 , d 4 > 0, µ 1 , µ 2 > 0, a 1 , a 2 > 0 and α, β > 0. The chemotactic function χ i (i = 1, 2) and the signal production function f i (i = 1, 2) are smooth. If n = 2, it is shown that this system possesses a unique global bounded classical solution provided that |χ i | (i = 1, 2) are bounded. If n ≤ 3, this system possesses a unique global bounded classical solution provided that µ i (i = 1, 2) are sufficiently large. Specifically, we first obtain an explicit formula µ i0 > 0 such that this system has no blow-up whenever µ i > µ i0 . Moreover, by constructing suitable energy functions, it is shown that:• If a 1 , a 2 ∈ (0, 1) and µ 1 and µ 2 are sufficiently large, then any global bounded solution exponentially converges to• If a 1 > 1 > a 2 > 0 and µ 2 is sufficiently large, then any global bounded solution exponentially converges to (0, f 1 (1)/α, 1, 0) as t → ∞;• If a 1 = 1 > a 2 > 0 and µ 2 is sufficiently large, then any global bounded solution algebraically converges to (0, f 1 (1)/α, 1, 0) as t → ∞. 2010 Mathematics Subject Classification. Primary: 35K35, 35A01, 35B44; Secondary: 35B35, 92C17.

show abstract

“…Taking into account, the results of the previous sections, ie, the functions

\underset{u}{true} false(t false)

and

\overset{u}{true} false(t false)

converge to u ∗ ( t ) as t → ∞ , where u ∗ ( t ) is a the periodic function defined in , we bound the solution of between

\underset{u}{true} false(t false)

(lower bound) and

\overset{u}{true} false(t false)

(upper bound) to obtain the same qualitative behavior than

\underset{u}{true} false(t false)

and

\overset{u}{true} false(t false)

. The proof follows the rectangle method used in Pao for reaction diffusion systems, see also Negreanu and Tello, where the method is applied to Parabolic‐Elliptic systems with chemotactic terms and to Negreanu and Tello …”

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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant

Cited by 89 publications

References 19 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

Boundedness and stabilization in a two-species chemotaxis system with two chemicals

On a parabolic‐elliptic chemotaxis system with periodic asymptotic behavior

Contact Info

Product

Resources

About