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January  2021, 26(1): 401-414. doi: 10.3934/dcdsb.2020083

On a singularly perturbed semi-linear problem with Robin boundary conditions

Qianqian Hou 1, , Tai-Chia Lin 2, and  Zhi-An Wang 3,

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

Received  September 2019 Published  January 2020

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition:
$ \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} ~~~~~~~~~~~~~~~~~~~~(*)$
where
$ \Omega \subset {\mathbb R}^N (N\geq 1) $
 is a bounded domain with smooth boundary,
$ \vec{n} $
 denotes the unit outward normal vector of
$ \partial \Omega $
 and
$ \gamma \in {\mathbb R}/\{0\} $
.
$ \varepsilon $
 and
$ \lambda $
 are positive constants. The problem (*) is derived from the well-known singular Keller-Segel system. When
$ \gamma>0 $
, we show there is only trivial solution
$ w = 0 $
. When
$ \gamma<0 $
 and
$ \Omega = B_R(0) $
 is a ball, we show that problem (*) has a non-constant solution which converges to zero uniformly as
$ \varepsilon $
 tends to zero. The main idea of this paper is to transform the Robin problem (*) to a nonlocal Dirichelt problem by a Cole-Hopf type transformation and then use the shooting method to obtain the existence of the transformed nonlocal Dirichlet problem. With the results for (*), we get the existence of non-constant stationary solutions to the original singular Keller-Segel system.
Keywords: Keller-Segel model, Robin boundary condition, nonlocal ODE, Cole-Hopf transformation, blowup radius.
Mathematics Subject Classification: 35A01, 35J15, 35J66, 35Q92, 92C17.
Citation: Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 401-414. doi: 10.3934/dcdsb.2020083
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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.  doi: 10.1007/BF00160334.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

T. Nagai and T. Ikeda, Traveling waves in a chemotactic model, J. Math. Biol., 30 (1991), 169-184.  doi: 10.1007/BF00160334.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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