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Boundedness in a two-species chemotaxis parabolic system with two chemicals
1. | College of Mathematic & Information, China West Normal University, Nanchong, 637002, China |
2. | School of Sciences, Southwest Petroleum University, Chengdu, 610500, China |
$\left\{\begin{aligned}&u_t=Δ u-χ\nabla·(u\nabla v), &x∈Ω,\,t>0,\\& τ v_t=Δ v-v+w, &x∈Ω,\,t>0,\\&w_t=Δ w-ξ\nabla·(w\nabla z), &x∈Ω,\,t>0,\\& τ z_t=Δ z-z+u, &x∈Ω,\,t>0,\end{aligned}\right.$ |
$χ, \, ξ∈\mathbb{R}$ |
$Ω\subset\mathbb{R}^2$ |
$\int_Ω u_0dx$ |
$\int_Ω w_0dx$ |
$τ=1$ |
References:
[1] |
N. D. Alikakos,
$L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
P. Biler, W. Hebisch and T. Nadzieja,
The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[3] |
P. Biler, E. E. Espejo and I. Guerra,
Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.
doi: 10.3934/cpaa.2013.12.89. |
[4] |
V. Calvez and J. A. Carrillo,
Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl., 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[5] |
T. Cieslak, P. LaurenÇot and C. Morales-Rodrigo,
Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117.
doi: 10.4064/bc81-0-7. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
![]() |
[7] |
M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and chemorepulsive cues direct the formation of the nigro-striatal circuit, Euro. J. Neurosci., 19 (2004), 831-844. Google Scholar |
[8] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.
|
[9] |
M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson,
Bacterial competition: surviving and thriving in the microbial jungle, Nature Reviews Microbiology., 8 (2010), 15-25.
doi: 10.1038/nrmicro2259. |
[10] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165.
|
[12] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[13] |
H. Jin and Z. A. Wang,
Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457.
doi: 10.1002/mma.3080. |
[14] |
H. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 206 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[15] |
H. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[16] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[17] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[18] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimers disease senile plague: Is there a connection?, Bull.Math. Biol., 65 (2003), 673-730. Google Scholar |
[19] |
X. Li,
Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301.
doi: 10.1002/mma.3477. |
[20] |
X. Li and Z. Xiang,
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Disc. Cont. Dyn. System -A, 35 (2015), 3503-3531.
doi: 10.3934/dcds.2015.35.3503. |
[21] |
X. Li and Z. Xiang,
On an attraction-repulsion chemotaxis system with logistic source, IMA J. of Appl. Math., 81 (2016), 165-198.
doi: 10.1093/imamat/hxv033. |
[22] |
K. Lin, C. Mu and L. Wang,
Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.
doi: 10.1016/j.jmaa.2014.12.052. |
[23] |
T. Nagai,
Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[24] |
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. |
[25] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A: Theory Methods and Applications, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[26] |
K. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.
|
[27] |
K. J. Painter and J. A. Sherratt,
Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[28] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[29] |
Y. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[30] |
Y. Tao,
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705-2722.
doi: 10.3934/dcdsb.2013.18.2705. |
[31] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[32] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[33] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[34] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discr. Cont. Dyn. Syst. B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[35] |
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. |
[36] |
Y. Wang,
A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292.
doi: 10.1016/j.jmaa.2016.03.061. |
[37] |
Y. Wang,
Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Comput. Math. Appl., 72 (2016), 2226-2240.
doi: 10.1016/j.camwa.2016.08.024. |
[38] |
Y. Wang,
Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22.
doi: 10.1186/s13661-016-0518-6. |
[39] |
Y. Wang and Z. Xiang,
Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discr. Cont. Dyn. Syst. B, 21 (2016), 1953-1973.
doi: 10.3934/dcdsb.2016031. |
[40] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[41] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[42] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[43] |
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] |
N. D. Alikakos,
$L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 10.1080/03605307908820113. |
[2] |
P. Biler, W. Hebisch and T. Nadzieja,
The Debye system: Existence and large time behavior of solutions, Nonlinear Anal., 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[3] |
P. Biler, E. E. Espejo and I. Guerra,
Blowup in higher dimensional two species chemotactic systems, Commun. Pure Appl. Anal., 12 (2013), 89-98.
doi: 10.3934/cpaa.2013.12.89. |
[4] |
V. Calvez and J. A. Carrillo,
Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pure Appl., 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[5] |
T. Cieslak, P. LaurenÇot and C. Morales-Rodrigo,
Global existence and convergence to steady-states in a chemorepulsion system, Banach Center Publications, 81 (2008), 105-117.
doi: 10.4064/bc81-0-7. |
[6] |
A. Friedman, Partial Differential Equations, Holt, Rinehart and Winston, New York, 1969.
![]() |
[7] |
M. A. Gates, V. M. Coupe, E. M. Torres, R. A. Fricker-Gares and S. B. Dunnett, Spatially and temporally restricted chemoattractant and chemorepulsive cues direct the formation of the nigro-striatal circuit, Euro. J. Neurosci., 19 (2004), 831-844. Google Scholar |
[8] |
M. A. Herrero and J. J. L. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Normale Superiore, 24 (1997), 633-683.
|
[9] |
M. E. Hibbing, C. Fuqua, M. R. Parsek and S. B. Peterson,
Bacterial competition: surviving and thriving in the microbial jungle, Nature Reviews Microbiology., 8 (2010), 15-25.
doi: 10.1038/nrmicro2259. |
[10] |
D. Horstmann and G. Wang,
Blow-up in a chemotaxis model without symmetry assumptions, European J. Appl. Math., 12 (2001), 159-177.
doi: 10.1017/S0956792501004363. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences, Ⅰ, Jahresber. Deutsch. Math.-Verien, 105 (2003), 103-165.
|
[12] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[13] |
H. Jin and Z. A. Wang,
Asymptotic dynamics of the one-dimensional attraction-repulsion Keller-Segel model, Math. Meth. Appl. Sci., 38 (2015), 444-457.
doi: 10.1002/mma.3080. |
[14] |
H. Jin and Z. A. Wang,
Boundedness, blowup and critical mass phenomenon in competing chemotaxis, J. Differential Equations, 206 (2016), 162-196.
doi: 10.1016/j.jde.2015.08.040. |
[15] |
H. Jin,
Boundedness of the attraction-repulsion Keller-Segel system, J. Math. Anal. Appl., 422 (2015), 1463-1478.
doi: 10.1016/j.jmaa.2014.09.049. |
[16] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[17] |
R. Kowalczyk and Z. Szymańska,
On the global existence of solutions to an aggregation model, J. Math. Anal. Appl., 343 (2008), 379-398.
doi: 10.1016/j.jmaa.2008.01.005. |
[18] |
M. Luca, A. Chavez-Ross, L. Edelstein-Keshet and A. Mogilner, Chemotactic signalling, microglia, and Alzheimers disease senile plague: Is there a connection?, Bull.Math. Biol., 65 (2003), 673-730. Google Scholar |
[19] |
X. Li,
Boundedness in a two-dimensional attraction-repulsion system with nonlinear diffusion, Math. Methods Appl. Sci., 39 (2016), 289-301.
doi: 10.1002/mma.3477. |
[20] |
X. Li and Z. Xiang,
Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source, Disc. Cont. Dyn. System -A, 35 (2015), 3503-3531.
doi: 10.3934/dcds.2015.35.3503. |
[21] |
X. Li and Z. Xiang,
On an attraction-repulsion chemotaxis system with logistic source, IMA J. of Appl. Math., 81 (2016), 165-198.
doi: 10.1093/imamat/hxv033. |
[22] |
K. Lin, C. Mu and L. Wang,
Large-time behavior of an attraction-repulsion chemotaxis system, J. Math. Anal. Appl., 426 (2015), 105-124.
doi: 10.1016/j.jmaa.2014.12.052. |
[23] |
T. Nagai,
Blow-up of nonradial solutions to parabolic-elliptic systems modelling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55.
doi: 10.1155/S1025583401000042. |
[24] |
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. |
[25] |
K. Osaki, T. Tsujikawa, A. Yagi and M. Mimura,
Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal. Ser. A: Theory Methods and Applications, 51 (2002), 119-144.
doi: 10.1016/S0362-546X(01)00815-X. |
[26] |
K. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Canad. Appl. Math. Quart., 10 (2002), 501-543.
|
[27] |
K. J. Painter and J. A. Sherratt,
Modelling the movement of interacting cell populations, Journal of Theoretical Biology, 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[28] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biology, 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[29] |
Y. Tao and Z. A. Wang,
Competing effects of attraction vs. repulsion in chemotaxis, Math. Models Methods Appl. Sci., 23 (2013), 1-36.
doi: 10.1142/S0218202512500443. |
[30] |
Y. Tao,
Global dynamics in a higher-dimensional repulsion chemotaxis model with nonlinear sensitivity, Discr. Cont. Dyn. Syst. B, 18 (2013), 2705-2722.
doi: 10.3934/dcdsb.2013.18.2705. |
[31] |
Y. Tao and M. Winkler,
A chemotaxis-haptotaxis model: The roles of nonlinear diffusion and logistic source, SIAM J. Math. Anal., 43 (2011), 685-704.
doi: 10.1137/100802943. |
[32] |
Y. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[33] |
Y. Tao and M. Winkler,
Energy-type estimates and global solvability in a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant, J. Differential Equations, 257 (2014), 784-815.
doi: 10.1016/j.jde.2014.04.014. |
[34] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discr. Cont. Dyn. Syst. B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[35] |
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. |
[36] |
Y. Wang,
A quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type with logistic source, J. Math. Anal. Appl., 441 (2016), 259-292.
doi: 10.1016/j.jmaa.2016.03.061. |
[37] |
Y. Wang,
Global bounded weak solutions to a degenerate quasilinear attraction-repulsion chemotaxis system with rotation, Comput. Math. Appl., 72 (2016), 2226-2240.
doi: 10.1016/j.camwa.2016.08.024. |
[38] |
Y. Wang,
Global existence and boundedness in a quasilinear attraction-repulsion chemotaxis system of parabolic-elliptic type, Bound. Value Probl., 2016 (2016), 1-22.
doi: 10.1186/s13661-016-0518-6. |
[39] |
Y. Wang and Z. Xiang,
Boundedness in a quasilinear 2D parabolic-parabolic attraction-repulsion chemotaxis system, Discr. Cont. Dyn. Syst. B, 21 (2016), 1953-1973.
doi: 10.3934/dcdsb.2016031. |
[40] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[41] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[42] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Commun. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[43] |
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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