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Decay in chemotaxis systems with a logistic term
Improvement of conditions for asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity
Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan |
$\begin{equation*} \begin{cases} u_t = d_1Δ u - \nabla · (u χ_1(w)\nabla w) +μ_1 u(1-u-a_1 v)&{\rm in} \ Ω × (0, ∞), \\ v_t = d_2Δ v - \nabla · (v χ_2(w)\nabla w) +μ_2 v(1-a_2u-v)&{\rm in} \ Ω × (0, ∞), \\ w_t = d_3Δ w + α u + β v - γ w&{\rm in} \ Ω × (0, ∞), \end{cases} \end{equation*}$ |
$Ω$ |
$\mathbb{R}^n$ |
$\partial Ω$ |
$n≥ 2$ |
$χ_i$ |
$a_1, a_2∈ (0, 1)$ |
$a_1, a_2∈ (0, 1)$ |
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species
chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[3] |
K. Fujie,
Boundedness in a fully parabolic chemotaxis system with singular sensitivit, J. Math. Anal. Appl., 424 (2015), 675-684.
doi: 10.1016/j.jmaa.2014.11.045. |
[4] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[5] |
M. Mizukami,
Remarks on smallness of chemotactic effect for asymptotic stability in a twospecies chemotaxis system, AIMS Mathematics, 1 (2016), 156-164.
doi: 10.3934/Math.2016.3.156. |
[6] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition
model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.
doi: 10.3934/dcdsb.2017097. |
[7] |
M. Mizukami,
Boundedness and stabilization in a two-species chemotaxis-competition system
of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.
doi: 10.1002/mma.4607. |
[8] |
M. Mizukami and T. Yokota,
Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.
doi: 10.1016/j.jde.2016.05.008. |
[9] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[10] |
M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with
non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[11] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis
model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[12] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic
source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[13] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
show all references
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species
chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[3] |
K. Fujie,
Boundedness in a fully parabolic chemotaxis system with singular sensitivit, J. Math. Anal. Appl., 424 (2015), 675-684.
doi: 10.1016/j.jmaa.2014.11.045. |
[4] |
K. Lin, C. Mu and L. Wang,
Boundedness in a two-species chemotaxis system, Math. Methods Appl. Sci., 38 (2015), 5085-5096.
doi: 10.1002/mma.3429. |
[5] |
M. Mizukami,
Remarks on smallness of chemotactic effect for asymptotic stability in a twospecies chemotaxis system, AIMS Mathematics, 1 (2016), 156-164.
doi: 10.3934/Math.2016.3.156. |
[6] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition
model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.
doi: 10.3934/dcdsb.2017097. |
[7] |
M. Mizukami,
Boundedness and stabilization in a two-species chemotaxis-competition system
of parabolic-parabolic-elliptic type, Math. Methods Appl. Sci., 41 (2018), 234-249.
doi: 10.1002/mma.4607. |
[8] |
M. Mizukami and T. Yokota,
Global existence and asymptotic stability of solutions to a twospecies chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.
doi: 10.1016/j.jde.2016.05.008. |
[9] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[10] |
M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with
non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[11] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis
model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[12] |
J. I. Tello and M. Winkler,
Stabilization in a two-species chemotaxis system with a logistic
source, Nonlinearity, 25 (2012), 1413-1425.
doi: 10.1088/0951-7715/25/5/1413. |
[13] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
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