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

Qianqian Hou; Tai-Chia Lin; Zhi-An Wang

doi:10.3934/dcdsb.2020083

Previous Article
The dynamics of a two host-two virus system in a chemostat environment
DCDS-B Home
This Issue
Next Article
Ecological and evolutionary dynamics in advective environments: Critical domain size and boundary conditions

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 2021 Early access 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
[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

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

[1]	Tian Xiang. On effects of sampling radius for the nonlocal Patlak-Keller-Segel chemotaxis model. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4911-4946. doi: 10.3934/dcds.2014.34.4911
[2]	Wenting Cong, Jian-Guo Liu. A degenerate $p$-Laplacian Keller-Segel model. Kinetic & Related Models, 2016, 9 (4) : 687-714. doi: 10.3934/krm.2016012
[3]	Hongyun Peng, Zhi-An Wang, Kun Zhao, Changjiang Zhu. Boundary layers and stabilization of the singular Keller-Segel system. Kinetic & Related Models, 2018, 11 (5) : 1085-1123. doi: 10.3934/krm.2018042
[4]	Qi Wang. Boundary spikes of a Keller-Segel chemotaxis system with saturated logarithmic sensitivity. Discrete & Continuous Dynamical Systems - B, 2015, 20 (4) : 1231-1250. doi: 10.3934/dcdsb.2015.20.1231
[5]	Shangbing Ai, Zhian Wang. Traveling bands for the Keller-Segel model with population growth. Mathematical Biosciences & Engineering, 2015, 12 (4) : 717-737. doi: 10.3934/mbe.2015.12.717
[6]	Vincent Calvez, Benoȋt Perthame, Shugo Yasuda. Traveling wave and aggregation in a flux-limited Keller-Segel model. Kinetic & Related Models, 2018, 11 (4) : 891-909. doi: 10.3934/krm.2018035
[7]	Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete & Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
[8]	Yadong Shang, Jianjun Paul Tian, Bixiang Wang. Asymptotic behavior of the stochastic Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1367-1391. doi: 10.3934/dcdsb.2019020
[9]	Yoshifumi Mimura. Critical mass of degenerate Keller-Segel system with no-flux and Neumann boundary conditions. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1603-1630. doi: 10.3934/dcds.2017066
[10]	Jiahang Che, Li Chen, Simone GÖttlich, Anamika Pandey, Jing Wang. Boundary layer analysis from the Keller-Segel system to the aggregation system in one space dimension. Communications on Pure & Applied Analysis, 2017, 16 (3) : 1013-1036. doi: 10.3934/cpaa.2017049
[11]	Tohru Tsujikawa, Kousuke Kuto, Yasuhito Miyamoto, Hirofumi Izuhara. Stationary solutions for some shadow system of the Keller-Segel model with logistic growth. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 1023-1034. doi: 10.3934/dcdss.2015.8.1023
[12]	Jean Dolbeault, Christian Schmeiser. The two-dimensional Keller-Segel model after blow-up. Discrete & Continuous Dynamical Systems, 2009, 25 (1) : 109-121. doi: 10.3934/dcds.2009.25.109
[13]	Shen Bian, Jian-Guo Liu, Chen Zou. Ultra-contractivity for Keller-Segel model with diffusion exponent $m>1-2/d$. Kinetic & Related Models, 2014, 7 (1) : 9-28. doi: 10.3934/krm.2014.7.9
[14]	Wenting Cong, Jian-Guo Liu. Uniform $L^{∞}$ boundedness for a degenerate parabolic-parabolic Keller-Segel model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 307-338. doi: 10.3934/dcdsb.2017015
[15]	Xinru Cao. Large time behavior in the logistic Keller-Segel model via maximal Sobolev regularity. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3369-3378. doi: 10.3934/dcdsb.2017141
[16]	Xiaofei Cao, Guowei Dai. Stability analysis of a model on varying domain with the Robin boundary condition. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 935-942. doi: 10.3934/dcdss.2017048
[17]	Qi Wang, Jingyue Yang, Lu Zhang. Time-periodic and stable patterns of a two-competing-species Keller-Segel chemotaxis model: Effect of cellular growth. Discrete & Continuous Dynamical Systems - B, 2017, 22 (9) : 3547-3574. doi: 10.3934/dcdsb.2017179
[18]	Ping Liu, Junping Shi, Zhi-An Wang. Pattern formation of the attraction-repulsion Keller-Segel system. Discrete & Continuous Dynamical Systems - B, 2013, 18 (10) : 2597-2625. doi: 10.3934/dcdsb.2013.18.2597
[19]	Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete & Continuous Dynamical Systems, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061
[20]	Xie Li, Zhaoyin Xiang. Boundedness in quasilinear Keller-Segel equations with nonlinear sensitivity and logistic source. Discrete & Continuous Dynamical Systems, 2015, 35 (8) : 3503-3531. doi: 10.3934/dcds.2015.35.3503

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors