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Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem
Global boundedness of solutions to the two-dimensional forager-exploiter model with logistic source
1. | College of Mathematics and Statistics, Chongqing University, Chongqing 401331, China |
2. | College of Mathematics and Statistics, Yili Normal University, Yining 835000, China |
$ \begin{equation*} \left\{ \begin{array}{lll} u_t = \Delta u- \nabla\cdot(u\nabla w) + \mu_1 (u-u^m), &x \in \Omega, t>0,\\ v_t = \Delta v - \nabla\cdot(v\nabla u) + \mu_2 ( v-v^l), &x\in \Omega, t>0,\\ w_t = \Delta w - \lambda(u+v)w - \mu w + r(x,t), & x\in \Omega, t>0, \end{array} \right. \end{equation*} $ |
$ \Omega \subset R^2 $ |
$ \mu $ |
$ \mu_1 $ |
$ \mu_2 $ |
$ \lambda $ |
$ m $ |
$ l $ |
$ 2\leq m < 3 $ |
$ l \geq 3 $ |
$ r(x,t) \in C^1(\overline{\Omega}\times[0,\infty))\cup L^{\infty}(\Omega\times(0,\infty)) $ |
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, in: Teubner-Texte Math., 133 1993, 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[2] |
T. Black,
Global generalized solutions to a forager-exploiter model with superlinear degradation and theri eventual regularity properties, Math. Models Methods Appl. Sci., 30 (2020), 1075-1117.
doi: 10.1142/S0218202520400072. |
[3] |
X. Cao,
Global radial renormalized solution to a producer-scrounger model with singular sensitivities, Math. Models Methods Appl. Sci., 6 (2020), 1119-1165.
doi: 10.1142/S0218202520400084. |
[4] |
H. Chen, J.-M. Li and K. Wang,
On the vanishing viscosity limit of a chemotaxis model, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1963-1987.
doi: 10.3934/dcds.2020101. |
[5] |
A. Friedman, Partial Different Equations, Holt, Rinehart and Winston, New York, 1969. Google Scholar |
[6] |
Y. Giga and H. Sohr,
Abstrat $L^p$ estimates for the Cauchy problem with aaplications to the Navier-Sotkes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[7] |
B. Hu and Y. Tao,
To the exclusion of blow-up in three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.
doi: 10.1142/S0218202516400091. |
[8] |
C. Jin,
Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 3547-3566.
doi: 10.3934/dcds.2018150. |
[9] |
H.-Y. Jin and Z.-A. Wang,
Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 3509-3527.
doi: 10.3934/dcds.2020027. |
[10] |
J. Lankeit and Y. Wang,
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 6099-6121.
doi: 10.3934/dcds.2017262. |
[11] |
H. Li and Y. Tao,
Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.
doi: 10.1016/j.aml.2017.10.006. |
[12] |
L. Meng, J. Yuan and X. Zheng,
Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 3413-3441.
doi: 10.3934/dcds.2019141. |
[13] |
N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[14] |
C. Mu, L. Wang, P. Zheng and Q. Zhang,
Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal.: Real World Appl., 14 (2013), 1634-1642.
doi: 10.1016/j.nonrwa.2012.10.022. |
[15] |
N. Tania, B. Vanderlei, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dunamic patterns of resource pathches and forager aggregation, Proc. Natl. Acad. Sci. USA, 109 (2012), 11228-11233. Google Scholar |
[16] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl, 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[17] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of larege-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[18] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[19] |
Y. Tao and M. Winkler,
Large time behavior in a forager-exploiter model with differnet taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.
doi: 10.1142/S021820251950043X. |
[20] |
J. Wang and M. Wang, Global bounded solution of the higher-dimensional forager-exploixer modle with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323.
doi: 10.1142/S0218202520500232. |
[21] |
H. Wang and Y. Li,
Boundedness in prey-taxis system with rotational flux terms, Commun. pur Appl.Anal, 19 (2020), 4839-4851.
doi: 10.3934/cpaa.2020214. |
[22] |
M. Winkler,
Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci., 29 (2019), 373-418.
doi: 10.1142/S021820251950012X. |
[23] |
M. Winkler,
Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2950.
doi: 10.1016/j.jde.2010.02.008. |
show all references
References:
[1] |
H. Amann, Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, Function Spaces, Differential Operators and Nonlinear Analysis, in: Teubner-Texte Math., 133 1993, 9-126.
doi: 10.1007/978-3-663-11336-2_1. |
[2] |
T. Black,
Global generalized solutions to a forager-exploiter model with superlinear degradation and theri eventual regularity properties, Math. Models Methods Appl. Sci., 30 (2020), 1075-1117.
doi: 10.1142/S0218202520400072. |
[3] |
X. Cao,
Global radial renormalized solution to a producer-scrounger model with singular sensitivities, Math. Models Methods Appl. Sci., 6 (2020), 1119-1165.
doi: 10.1142/S0218202520400084. |
[4] |
H. Chen, J.-M. Li and K. Wang,
On the vanishing viscosity limit of a chemotaxis model, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 1963-1987.
doi: 10.3934/dcds.2020101. |
[5] |
A. Friedman, Partial Different Equations, Holt, Rinehart and Winston, New York, 1969. Google Scholar |
[6] |
Y. Giga and H. Sohr,
Abstrat $L^p$ estimates for the Cauchy problem with aaplications to the Navier-Sotkes equations in exterior domains, J. Funct. Anal., 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[7] |
B. Hu and Y. Tao,
To the exclusion of blow-up in three-dimensional chemotaxis-growth model with indirect attractant production, Math. Models Methods Appl. Sci., 26 (2016), 2111-2128.
doi: 10.1142/S0218202516400091. |
[8] |
C. Jin,
Global classical solution and stability to a coupled chemotaxis-fluid model with logistic source, Discrete Contin. Dyn. Syst. Ser. A, 38 (2018), 3547-3566.
doi: 10.3934/dcds.2018150. |
[9] |
H.-Y. Jin and Z.-A. Wang,
Global stabilization of the full attraction-repulsion Keller-Segel system, Discrete Contin. Dyn. Syst. Ser. A, 40 (2020), 3509-3527.
doi: 10.3934/dcds.2020027. |
[10] |
J. Lankeit and Y. Wang,
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst. Ser. A, 37 (2017), 6099-6121.
doi: 10.3934/dcds.2017262. |
[11] |
H. Li and Y. Tao,
Boundedness in a chemotaxis system with indirect signal production and generalized logistic source, Appl. Math. Lett., 77 (2018), 108-113.
doi: 10.1016/j.aml.2017.10.006. |
[12] |
L. Meng, J. Yuan and X. Zheng,
Global existence of almost energy solution to the two-dimensional chemotaxis-Navier-Stokes equations with partial diffusion, Discrete Contin. Dyn. Syst. Ser. A, 39 (2019), 3413-3441.
doi: 10.3934/dcds.2019141. |
[13] |
N. Mizoguchi and P. Souplet, Nondegeneracy of blow-up points for the parabolic Keller-Segel system, Ann. Inst. H. Poincar$\acute{e}$ Anal. Non Lin$\acute{e}$aire, 31 (2014), 851-875.
doi: 10.1016/j.anihpc.2013.07.007. |
[14] |
C. Mu, L. Wang, P. Zheng and Q. Zhang,
Global existence and boundedness of classical solutions to a parabolic-parabolic chemotaxis system, Nonlinear Anal.: Real World Appl., 14 (2013), 1634-1642.
doi: 10.1016/j.nonrwa.2012.10.022. |
[15] |
N. Tania, B. Vanderlei, J. P. Heath and L. Edelstein-Keshet, Role of social interactions in dunamic patterns of resource pathches and forager aggregation, Proc. Natl. Acad. Sci. USA, 109 (2012), 11228-11233. Google Scholar |
[16] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl, 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[17] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of larege-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[18] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[19] |
Y. Tao and M. Winkler,
Large time behavior in a forager-exploiter model with differnet taxis strategies for two groups in search of food, Math. Models Methods Appl. Sci., 29 (2019), 2151-2182.
doi: 10.1142/S021820251950043X. |
[20] |
J. Wang and M. Wang, Global bounded solution of the higher-dimensional forager-exploixer modle with/without growth sources, Math. Models Methods Appl. Sci., 30 (2020), 1297-1323.
doi: 10.1142/S0218202520500232. |
[21] |
H. Wang and Y. Li,
Boundedness in prey-taxis system with rotational flux terms, Commun. pur Appl.Anal, 19 (2020), 4839-4851.
doi: 10.3934/cpaa.2020214. |
[22] |
M. Winkler,
Global generalized solutions to a multi-dimensional doubly tactic resource consumption model accounting for social interactions, Math. Models Methods Appl. Sci., 29 (2019), 373-418.
doi: 10.1142/S021820251950012X. |
[23] |
M. Winkler,
Aggregation vs.global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2950.
doi: 10.1016/j.jde.2010.02.008. |
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