Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation

Previous Article
Resurgence of inner solutions for perturbations of the McMillan map
DCDS Home
This Issue
Next Article
Global attractors for strongly damped wave equations with displacement dependent damping and nonlinear source term of critical exponent

March 2011, 31(1): 139-164. doi: 10.3934/dcds.2011.31.139

Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation

Kazuhiro Kurata ^1, and Kotaro Morimoto ^1,

1.	Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, Japan

Received January 2010 Revised March 2011 Published June 2011

Abstract
Related Papers

We are concerned with a multiple boundary spike solution to the steady-state problem of a chemotaxis system: $P_t=\nabla \cdot \big( P\nabla ( \log \frac{P}{\Phi (W)})\big)$, $W_t=ε^2 \Delta W+F(P,W)$, in $\Omega \times (0,\infty)$, under the homogeneous Neumann boundary condition, where $\Omega\subset \mathbb{R}^N$ is a bounded domain with smooth boundary, $P(x,t)$ is a population density, $W(x,t)$ is a density of chemotaxis substance. We assume that $\Phi(W)=W^p$, $p>1$, and we are interested in the cases of $F(P,W)=-W+\frac{PW^q}{\alpha+\gamma W^q}$ and $F(P,W)=-W+\frac{P}{1+ k P}$ with $q>0, \alpha, \gamma, k\ge 0$, which has a saturating growth. Existence of a multiple spike stationary pattern is related to a weak saturation effect of $F(P,W)$ and the shape of the domain $\Omega$. In this paper, we assume that $\Omega$ is symmetric with respect to each hyperplane $\{ x_1=0\},\cdots ,\{ x_{N-1}=0\}$. For two classes of $F(P,W)$ above with saturation effect, we show the existence of multiple boundary spike stationary patterns on $\Omega$ under a weak saturation effect on parameters $\alpha,\gamma$ and $k$. Based on the method developed in [14] and [10], we shall present some technique to construct a multiple boundary spike solution to some reduced nonlocal problem on such domains systematically.

Keywords: chemotaxis, saturation effect., spike patterns, Nonlinear elliptic system.

Mathematics Subject Classification: Primary: 35K50, 35Q80; Secondary: 92C1.

Citation: Kazuhiro Kurata, Kotaro Morimoto. Existence of multiple spike stationary patterns in a chemotaxis model with weak saturation. Discrete and Continuous Dynamical Systems, 2011, 31 (1) : 139-164. doi: 10.3934/dcds.2011.31.139

References:

[1]	P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations, 4 (1999), 1-69.
[2]	P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7.
[3]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-145. doi: 10.1007/BF00250555.
[4]	H. Berestycki, T. Gallouët and O. Kavian, Nonlinear Euclidean scalar field equations in the plane, C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310.
[5]	M. A. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837. doi: 10.2307/2155064.
[6]	P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232. doi: 10.1007/BF00247733.
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
[8]	D. Iron, M. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62. doi: 10.1016/S0167-2789(00)00206-2.
[9]	T. Kolokolnikov, W. Sun, M. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Syst., 5 (2006), 313-363. doi: 10.1137/050635080.
[10]	K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation, Commun. Pure Appl. Anal., 7 (2008), 1443-1482. doi: 10.3934/cpaa.2008.7.1443.
[11]	M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363. doi: 10.2307/2154113.
[12]	M. K. Kwong and L. Q. Zhang, Uniqueness of the positive solutions of $\Delta u+f(u)=0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599.
[13]	H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106.
[14]	W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365. doi: 10.1007/BF03167294.
[15]	W.-M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6.
[16]	H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.
[17]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems. II, J. Differential Equations, 158 (1999), 94-151. doi: 10.1016/S0022-0396(99)80020-5.
[18]	X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Appl. Math., 70 (2009), 1120-1138. doi: 10.1137/080742361.
[19]	K. Sakamoto, Internal layers in high-dimensional domains, Proc. Roy. Soc. Edinburgh Sect. A., 128 (1998), 359-401.
[20]	T. Senba and T. Suzuki, "Applied Analysis. Mathematical Methods in Natural Science," Imperial College Press, London, 2004.
[21]	B. D. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117.
[22]	T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkäuser Boston, Inc., Boston, MA, 2005.
[23]	J. Wei, "Existence and Stability of Spikes for the Gierer-Meinhardt System," Handbook of Differential Equations: Stationary Partial Differential Equations, V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 487-585. doi: 10.1016/S1874-5733(08)80013-7.
[24]	J. Wei and M. Winter, On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal., 30 (1999), 1241-1263. doi: 10.1137/S0036141098347237.
[25]	J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518. doi: 10.1006/jdeq.2001.4019.
[26]	J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277. doi: 10.1142/S021919970400132X.
[27]	J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $R^1$, Methods Appl. Anal., 14 (2007), 119-163.
[28]	J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y.
[29]	E. Zeidler, "Nonlinear Functional Analysis and its Applications. I, Fixed-Point Theorems," Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986.

show all references

References:

[1]	P. Bates, E. N. Dancer and J. Shi, Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability, Adv. Differential Equations, 4 (1999), 1-69.
[2]	P. Bates and J. Shi, Existence and instability of spike layer solutions to singular perturbation problems, J. Funct. Anal., 196 (2002), 211-264. doi: 10.1016/S0022-1236(02)00013-7.
[3]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-145. doi: 10.1007/BF00250555.
[4]	H. Berestycki, T. Gallouët and O. Kavian, Nonlinear Euclidean scalar field equations in the plane, C. R. Acad. Sci. Paris Sér. I Math., 297 (1983), 307-310.
[5]	M. A. del Pino, Radially symmetric internal layers in a semilinear elliptic system, Trans. Amer. Math. Soc., 347 (1995), 4807-4837. doi: 10.2307/2155064.
[6]	P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232. doi: 10.1007/BF00247733.
[7]	D. Horstmann, From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
[8]	D. Iron, M. Ward and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D, 150 (2001), 25-62. doi: 10.1016/S0167-2789(00)00206-2.
[9]	T. Kolokolnikov, W. Sun, M. Ward and J. Wei, The stability of a stripe for the Gierer-Meinhardt model and the effect of saturation, SIAM J. Appl. Dyn. Syst., 5 (2006), 313-363. doi: 10.1137/050635080.
[10]	K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation, Commun. Pure Appl. Anal., 7 (2008), 1443-1482. doi: 10.3934/cpaa.2008.7.1443.
[11]	M. K. Kwong and Y. Li, Uniqueness of radial solutions of semilinear elliptic equations, Trans. Amer. Math. Soc., 333 (1992), 339-363. doi: 10.2307/2154113.
[12]	M. K. Kwong and L. Q. Zhang, Uniqueness of the positive solutions of $\Delta u+f(u)=0$ in an annulus, Differential Integral Equations, 4 (1991), 583-599.
[13]	H. A. Levine and B. D. Sleeman, A system of reaction diffusion equation arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730. doi: 10.1137/S0036139995291106.
[14]	W.-M. Ni and I. Takagi, Point condensation generated by a reaction-diffusion system in axially symmetric domains, Japan J. Indust. Appl. Math., 12 (1995), 327-365. doi: 10.1007/BF03167294.
[15]	W.-M. Ni, Qualitative properties of solutions to elliptic problems, Stationary Partial Differential Equations, I, Handb. Differ. Equ., North-Holland, Amsterdam, (2004), 157-233. doi: 10.1016/S1874-5733(04)80005-6.
[16]	H. G. Othmer and A. Stevens, Aggregation, blowup, and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081. doi: 10.1137/S0036139995288976.
[17]	T. Ouyang and J. Shi, Exact multiplicity of positive solutions for a class of semilinear problems. II, J. Differential Equations, 158 (1999), 94-151. doi: 10.1016/S0022-0396(99)80020-5.
[18]	X. Ren and J. Wei, Oval shaped droplet solutions in the saturation process of some pattern formation problems, SIAM J. Appl. Math., 70 (2009), 1120-1138. doi: 10.1137/080742361.
[19]	K. Sakamoto, Internal layers in high-dimensional domains, Proc. Roy. Soc. Edinburgh Sect. A., 128 (1998), 359-401.
[20]	T. Senba and T. Suzuki, "Applied Analysis. Mathematical Methods in Natural Science," Imperial College Press, London, 2004.
[21]	B. D. Sleeman, M. J. Ward and J. Wei, The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl Math., 65 (2005), 790-817. doi: 10.1137/S0036139902415117.
[22]	T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkäuser Boston, Inc., Boston, MA, 2005.
[23]	J. Wei, "Existence and Stability of Spikes for the Gierer-Meinhardt System," Handbook of Differential Equations: Stationary Partial Differential Equations, V, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, (2008), 487-585. doi: 10.1016/S1874-5733(08)80013-7.
[24]	J. Wei and M. Winter, On the two-dimensional Gierer-Meinhardt system with strong coupling, SIAM J. Math. Anal., 30 (1999), 1241-1263. doi: 10.1137/S0036141098347237.
[25]	J. Wei and M. Winter, Spikes for the Gierer-Meinhardt system in two dimensions: The strong coupling case, J. Differential Equations, 178 (2002), 478-518. doi: 10.1006/jdeq.2001.4019.
[26]	J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277. doi: 10.1142/S021919970400132X.
[27]	J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $R^1$, Methods Appl. Anal., 14 (2007), 119-163.
[28]	J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion systems, J. Math. Biol., 57 (2008), 53-89. doi: 10.1007/s00285-007-0146-y.
[29]	E. Zeidler, "Nonlinear Functional Analysis and its Applications. I, Fixed-Point Theorems," Translated from the German by Peter R. Wadsack, Springer-Verlag, New York, 1986.

[1]	Tian Xiang. Dynamics in a parabolic-elliptic chemotaxis system with growth source and nonlinear secretion. Communications on Pure and Applied Analysis, 2019, 18 (1) : 255-284. doi: 10.3934/cpaa.2019014
[2]	Yilong Wang, Xuande Zhang. On a parabolic-elliptic chemotaxis-growth system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 321-328. doi: 10.3934/dcdss.2020018
[3]	Youshan Tao, Michael Winkler. A chemotaxis-haptotaxis system with haptoattractant remodeling: Boundedness enforced by mild saturation of signal production. Communications on Pure and Applied Analysis, 2019, 18 (4) : 2047-2067. doi: 10.3934/cpaa.2019092
[4]	Hong Yi, Chunlai Mu, Shuyan Qiu, Lu Xu. Global boundedness of radial solutions to a parabolic-elliptic chemotaxis system with flux limitation and nonlinear signal production. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3825-3849. doi: 10.3934/cpaa.2021133
[5]	Yuya Tanaka, Tomomi Yokota. Finite-time blow-up in a quasilinear degenerate parabolic–elliptic chemotaxis system with logistic source and nonlinear production. Discrete and Continuous Dynamical Systems - B, 2023, 28 (1) : 262-286. doi: 10.3934/dcdsb.2022075
[6]	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 and Continuous Dynamical Systems - B, 2017, 22 (9) : 3547-3574. doi: 10.3934/dcdsb.2017179
[7]	M. Guedda, R. Kersner, M. Klincsik, E. Logak. Exact wavefronts and periodic patterns in a competition system with nonlinear diffusion. Discrete and Continuous Dynamical Systems - B, 2014, 19 (6) : 1589-1600. doi: 10.3934/dcdsb.2014.19.1589
[8]	Luis F. Gordillo, Stephen A. Marion, Priscilla E. Greenwood. The effect of patterns of infectiousness on epidemic size. Mathematical Biosciences & Engineering, 2008, 5 (3) : 429-435. doi: 10.3934/mbe.2008.5.429
[9]	Yūki Naito, Takasi Senba. Oscillating solutions to a parabolic-elliptic system related to a chemotaxis model. Conference Publications, 2011, 2011 (Special) : 1111-1118. doi: 10.3934/proc.2011.2011.1111
[10]	Muhammad Altaf Khan, Muhammad Farhan, Saeed Islam, Ebenezer Bonyah. Modeling the transmission dynamics of avian influenza with saturation and psychological effect. Discrete and Continuous Dynamical Systems - S, 2019, 12 (3) : 455-474. doi: 10.3934/dcdss.2019030
[11]	Weichung Wang, Tsung-Fang Wu, Chien-Hsiang Liu. On the multiple spike solutions for singularly perturbed elliptic systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (1) : 237-258. doi: 10.3934/dcdsb.2013.18.237
[12]	Pan Zheng, Chunlai Mu, Xuegang Hu. Boundedness and blow-up for a chemotaxis system with generalized volume-filling effect and logistic source. Discrete and Continuous Dynamical Systems, 2015, 35 (5) : 2299-2323. doi: 10.3934/dcds.2015.35.2299
[13]	Yajing Zhang, Xinfu Chen, Jianghao Hao, Xin Lai, Cong Qin. Dynamics of spike in a Keller-Segel's minimal chemotaxis model. Discrete and Continuous Dynamical Systems, 2017, 37 (2) : 1109-1127. doi: 10.3934/dcds.2017046
[14]	Yuanyuan Liu, Youshan Tao. Asymptotic behavior in a chemotaxis-growth system with nonlinear production of signals. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 465-475. doi: 10.3934/dcdsb.2017021
[15]	Pan Zheng, Chunlai Mu, Xiaojun Song. On the boundedness and decay of solutions for a chemotaxis-haptotaxis system with nonlinear diffusion. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1737-1757. doi: 10.3934/dcds.2016.36.1737
[16]	Runlin Hu, Pan Zheng. On a quasilinear fully parabolic attraction or repulsion chemotaxis system with nonlinear signal production. Discrete and Continuous Dynamical Systems - B, 2022, 27 (12) : 7227-7244. doi: 10.3934/dcdsb.2022041
[17]	Uchida Hidetake. Analytic smoothing effect and global existence of small solutions for the elliptic-hyperbolic Davey-Stewartson system. Conference Publications, 2001, 2001 (Special) : 182-190. doi: 10.3934/proc.2001.2001.182
[18]	Rachidi B. Salako, Wenxian Shen. Spreading speeds and traveling waves of a parabolic-elliptic chemotaxis system with logistic source on $\mathbb{R}^N$. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6189-6225. doi: 10.3934/dcds.2017268
[19]	Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009
[20]	Silvia Frassu, Rafael Rodríguez Galván, Giuseppe Viglialoro. Uniform in time $ L^\infty $-estimates for an attraction-repulsion chemotaxis model with double saturation. Discrete and Continuous Dynamical Systems - B, 2022 doi: 10.3934/dcdsb.2022151

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Kazuhiro Kurata Kotaro Morimoto
on Google Scholar
Kazuhiro Kurata Kotaro Morimoto

[Back to Top]