# American Institute of Mathematical Sciences

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

## 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

Received  September 2019 Published  January 2021 Early access  January 2020

This paper is concerned with a semi-linear elliptic problem with Robin boundary condition:
 $$$\left\{\begin{array}{lll} \varepsilon \Delta w-\lambda w^{1+\chi} = 0, &\text{in} \ \Omega\\ \nabla w \cdot \vec{n}+\gamma w = 0, & \text{on} \ \partial \Omega \end{array}\right.$$ ~~~~~~~~~~~~~~~~~~~~(*)$
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.
Citation: Qianqian Hou, Tai-Chia Lin, Zhi-An Wang. On a singularly perturbed semi-linear problem with Robin boundary conditions. Discrete and 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. [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. [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. [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. [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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##### References:
 [1] J. Adler, Chemotaxis in bacteria, Science, 153 (1966), 708-716. [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. [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. [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. [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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