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Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions
On a singularly perturbed semi-linear problem with Robin boundary conditions
1. | Institute for Advanced Study in Mathematics, Harbin Institute of Technology, Harbin 150001, China |
2. | Institute of Applied Mathematical Sciences and National Center for Theoretical Sciences (NCTS), National Taiwan University, Taipei 10617, Taiwan |
3. | Department of Applied Mathematics, Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China |
$ \begin{equation} \left\{\begin{array}{lll} \varepsilon \Delta w-\lambda w^{1+\chi} = 0, &\text{in} \ \Omega\\ \nabla w \cdot \vec{n}+\gamma w = 0, & \text{on} \ \partial \Omega \end{array}\right. \end{equation} ~~~~~~~~~~~~~~~~~~~~(*)$ |
$ \Omega \subset {\mathbb R}^N (N\geq 1) $ |
$ \vec{n} $ |
$ \partial \Omega $ |
$ \gamma \in {\mathbb R}/\{0\} $ |
$ \varepsilon $ |
$ \lambda $ |
$ \gamma>0 $ |
$ w = 0 $ |
$ \gamma<0 $ |
$ \Omega = B_R(0) $ |
$ \varepsilon $ |
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. Google Scholar |
[2] |
M. Chae and K. Choi, Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion, arXiv: 1903.04372v1. Google Scholar |
[3] |
M. Chae, K. Choi, K. Kang and J. Lee,
Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain, J. Differential Equations, 265 (2018), 237-279.
doi: 10.1016/j.jde.2018.02.034. |
[4] |
K. Choi, M.-J. Kang, Y.-S. Kwon and A. Vasseur, Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model, Mathematical Models and Methods in Applied Sciences, 2019, arXiv: 1904.12169v1.
doi: 10.1142/S0218202520500104. |
[5] |
P. Davis, P. van Heijster and R. Marangell,
Absolute instabilities of travelling wave solutions in a Keller-Segel model, Nonlinearity, 30 (2017), 4029-4061.
doi: 10.1088/1361-6544/aa842f. |
[6] |
C. Deng and T. Li,
Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.
doi: 10.1016/j.jde.2014.05.014. |
[7] |
C. Hao,
Global well-posedness for a multidimensional chemotaxis model in critical besov spaces, Z. Angew Math. Phys., 63 (2012), 825-834.
doi: 10.1007/s00033-012-0193-0. |
[8] |
Q. Hou, C. J. Liu, Y. G. Wang and Z.A. Wang,
Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: One dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.
doi: 10.1137/17M112748X. |
[9] |
Q. Hou and Z.A. Wang,
Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures. Appl., 130 (2019), 251-287.
doi: 10.1016/j.matpur.2019.01.008. |
[10] |
Q. Hou, Z.A. Wang and K. Zhao,
Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differential Equations, 261 (2016), 5035-5070.
doi: 10.1016/j.jde.2016.07.018. |
[11] |
H. Jin, J. Li and Z.A. Wang,
Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[12] |
Y. Kalinin, L. Jiang, Y. Tu and M. Wu,
Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[13] |
E. Keller and G. Odell,
Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1975), 309-317.
doi: 10.1016/0025-5564(75)90109-1. |
[14] |
E. Keller and L. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.
doi: 10.1016/0022-5193(71)90051-8. |
[15] |
H. Levine, B. Sleeman and M. Nilsen-Hamilton,
A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. Ⅰ. the role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 77-115.
doi: 10.1016/S0025-5564(00)00034-1. |
[16] |
D. Li, R. Pan and K. Zhao,
Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 28 (2015), 2181-2210.
doi: 10.1088/0951-7715/28/7/2181. |
[17] |
H. Li and K. Zhao,
Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338.
doi: 10.1016/j.jde.2014.09.014. |
[18] |
J. Li, T. Li and Z.A. Wang,
Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.
doi: 10.1142/S0218202514500389. |
[19] |
T. Li, R. Pan and K. Zhao,
Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[20] |
T. Li and Z.A. Wang,
Nonlinear stability of travelling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.
doi: 10.1137/09075161X. |
[21] |
T. Li and Z.A. Wang,
Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[22] |
R. Lui and Z.A. Wang,
Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[23] |
V. Martinez, Z.A. Wang and K. Zhao,
Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383-1424.
doi: 10.1512/iumj.2018.67.7394. |
[24] |
T. Nagai and T. Ikeda,
Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[25] |
R. Nossal, Boundary movement of chemotactic bacterial populations, Math. Biosci., 13 (1972), 397-406. Google Scholar |
[26] |
H. Peng, H. Wen and C. Zhu,
Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew Math. Phys., 65 (2014), 1167-1188.
doi: 10.1007/s00033-013-0378-1. |
[27] |
L. Rebholz, D. Wang, Z.A. Wang, C. Zerfas and K. Zhao,
Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Disc. Cont. Dyn. Syst., 39 (2019), 3789-3838.
doi: 10.3934/dcds.2019154. |
[28] |
H. Schwetlick, Traveling waves for chemotaxis–systems, in PAMM: Proceedings in Applied Mathematics and Mechanics, Wiley Online Library, 3 (2003), 476–478. Google Scholar |
[29] |
Z.A. Wang,
Mathematics of traveling waves in chemotaxis, Disc. Cont. Dyn. Syst.-Series B., 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[30] |
Z.A. Wang, Z. Xiang and P. Yu,
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.
doi: 10.1016/j.jde.2015.09.063. |
[31] |
M. Winkler,
Renormalized radial large-data solutions to the higher-dimensional keller-segel system with singular sensitivity and signal absorption, J. Differential Equations, 264 (2018), 2310-2350.
doi: 10.1016/j.jde.2017.10.029. |
show all references
References:
[1] |
J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. Google Scholar |
[2] |
M. Chae and K. Choi, Nonlinear stability of planar traveling waves in a chemotaxis model of tumor angiogenesis with chemical diffusion, arXiv: 1903.04372v1. Google Scholar |
[3] |
M. Chae, K. Choi, K. Kang and J. Lee,
Stability of planar traveling waves in a Keller-Segel equation on an infinite strip domain, J. Differential Equations, 265 (2018), 237-279.
doi: 10.1016/j.jde.2018.02.034. |
[4] |
K. Choi, M.-J. Kang, Y.-S. Kwon and A. Vasseur, Contraction for large perturbations of traveling waves in a hyperbolic-parabolic system arising from a chemotaxis model, Mathematical Models and Methods in Applied Sciences, 2019, arXiv: 1904.12169v1.
doi: 10.1142/S0218202520500104. |
[5] |
P. Davis, P. van Heijster and R. Marangell,
Absolute instabilities of travelling wave solutions in a Keller-Segel model, Nonlinearity, 30 (2017), 4029-4061.
doi: 10.1088/1361-6544/aa842f. |
[6] |
C. Deng and T. Li,
Well-posedness of a 3D parabolic-hyperbolic Keller-Segel system in the sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.
doi: 10.1016/j.jde.2014.05.014. |
[7] |
C. Hao,
Global well-posedness for a multidimensional chemotaxis model in critical besov spaces, Z. Angew Math. Phys., 63 (2012), 825-834.
doi: 10.1007/s00033-012-0193-0. |
[8] |
Q. Hou, C. J. Liu, Y. G. Wang and Z.A. Wang,
Stability of boundary layers for a viscous hyperbolic system arising from chemotaxis: One dimensional case, SIAM J. Math. Anal., 50 (2018), 3058-3091.
doi: 10.1137/17M112748X. |
[9] |
Q. Hou and Z.A. Wang,
Convergence of boundary layers for the Keller-Segel system with singular sensitivity in the half-plane, J. Math. Pures. Appl., 130 (2019), 251-287.
doi: 10.1016/j.matpur.2019.01.008. |
[10] |
Q. Hou, Z.A. Wang and K. Zhao,
Boundary layer problem on a hyperbolic system arising from chemotaxis, J. Differential Equations, 261 (2016), 5035-5070.
doi: 10.1016/j.jde.2016.07.018. |
[11] |
H. Jin, J. Li and Z.A. Wang,
Asymptotic stability of traveling waves of a chemotaxis model with singular sensitivity, J. Differential Equations, 255 (2013), 193-219.
doi: 10.1016/j.jde.2013.04.002. |
[12] |
Y. Kalinin, L. Jiang, Y. Tu and M. Wu,
Logarithmic sensing in Escherichia coli bacterial chemotaxis, Biophysical J., 96 (2009), 2439-2448.
doi: 10.1016/j.bpj.2008.10.027. |
[13] |
E. Keller and G. Odell,
Necessary and sufficient conditions for chemotactic bands, Math. Biosci., 27 (1975), 309-317.
doi: 10.1016/0025-5564(75)90109-1. |
[14] |
E. Keller and L. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 377-380.
doi: 10.1016/0022-5193(71)90051-8. |
[15] |
H. Levine, B. Sleeman and M. Nilsen-Hamilton,
A mathematical model for the roles of pericytes and macrophages in the initiation of angiogenesis. Ⅰ. the role of protease inhibitors in preventing angiogenesis, Math. Biosci., 168 (2000), 77-115.
doi: 10.1016/S0025-5564(00)00034-1. |
[16] |
D. Li, R. Pan and K. Zhao,
Quantitative decay of a one-dimensional hybrid chemotaxis model with large data, Nonlinearity, 28 (2015), 2181-2210.
doi: 10.1088/0951-7715/28/7/2181. |
[17] |
H. Li and K. Zhao,
Initial-boundary value problems for a system of hyperbolic balance laws arising from chemotaxis, J. Differential Equations, 258 (2015), 302-338.
doi: 10.1016/j.jde.2014.09.014. |
[18] |
J. Li, T. Li and Z.A. Wang,
Stability of traveling waves of the Keller-Segel system with logarithmic sensitivity, Math. Models Methods Appl. Sci., 24 (2014), 2819-2849.
doi: 10.1142/S0218202514500389. |
[19] |
T. Li, R. Pan and K. Zhao,
Global dynamics of a hyperbolic-parabolic model arising from chemotaxis, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[20] |
T. Li and Z.A. Wang,
Nonlinear stability of travelling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.
doi: 10.1137/09075161X. |
[21] |
T. Li and Z.A. Wang,
Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[22] |
R. Lui and Z.A. Wang,
Traveling wave solutions from microscopic to macroscopic chemotaxis models, J. Math. Biol., 61 (2010), 739-761.
doi: 10.1007/s00285-009-0317-0. |
[23] |
V. Martinez, Z.A. Wang and K. Zhao,
Asymptotic and viscous stability of large-amplitude solutions of a hyperbolic system arising from biology, Indiana Univ. Math. J., 67 (2018), 1383-1424.
doi: 10.1512/iumj.2018.67.7394. |
[24] |
T. Nagai and T. Ikeda,
Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[25] |
R. Nossal, Boundary movement of chemotactic bacterial populations, Math. Biosci., 13 (1972), 397-406. Google Scholar |
[26] |
H. Peng, H. Wen and C. Zhu,
Global well-posedness and zero diffusion limit of classical solutions to 3D conservation laws arising in chemotaxis, Z. Angew Math. Phys., 65 (2014), 1167-1188.
doi: 10.1007/s00033-013-0378-1. |
[27] |
L. Rebholz, D. Wang, Z.A. Wang, C. Zerfas and K. Zhao,
Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions, Disc. Cont. Dyn. Syst., 39 (2019), 3789-3838.
doi: 10.3934/dcds.2019154. |
[28] |
H. Schwetlick, Traveling waves for chemotaxis–systems, in PAMM: Proceedings in Applied Mathematics and Mechanics, Wiley Online Library, 3 (2003), 476–478. Google Scholar |
[29] |
Z.A. Wang,
Mathematics of traveling waves in chemotaxis, Disc. Cont. Dyn. Syst.-Series B., 18 (2013), 601-641.
doi: 10.3934/dcdsb.2013.18.601. |
[30] |
Z.A. Wang, Z. Xiang and P. Yu,
Asymptotic dynamics on a singular chemotaxis system modeling onset of tumor angiogenesis, J. Differential Equations, 260 (2016), 2225-2258.
doi: 10.1016/j.jde.2015.09.063. |
[31] |
M. Winkler,
Renormalized radial large-data solutions to the higher-dimensional keller-segel system with singular sensitivity and signal absorption, J. Differential Equations, 264 (2018), 2310-2350.
doi: 10.1016/j.jde.2017.10.029. |
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