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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232. doi: 10.1007/BF00247733.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. Sakamoto, Internal layers in high-dimensional domains, Proc. Roy. Soc. Edinburgh Sect. A., 128 (1998), 359-401.  Google Scholar

[20]

T. Senba and T. Suzuki, "Applied Analysis. Mathematical Methods in Natural Science," Imperial College Press, London, 2004.  Google Scholar

[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.  Google Scholar

[22]

T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkäuser Boston, Inc., Boston, MA, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277. doi: 10.1142/S021919970400132X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

P. C. Fife, Semilinear elliptic boundary value problems with small parameters, Arch. Rational Mech. Anal., 52 (1973), 205-232. doi: 10.1007/BF00247733.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[19]

K. Sakamoto, Internal layers in high-dimensional domains, Proc. Roy. Soc. Edinburgh Sect. A., 128 (1998), 359-401.  Google Scholar

[20]

T. Senba and T. Suzuki, "Applied Analysis. Mathematical Methods in Natural Science," Imperial College Press, London, 2004.  Google Scholar

[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.  Google Scholar

[22]

T. Suzuki, "Free Energy and Self-Interacting Particles," Progress in Nonlinear Differential Equations and their Applications, 62, Birkäuser Boston, Inc., Boston, MA, 2005.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. Wei and M. Winter, On the Gierer-Meinhardt system with saturation, Commun. Contemp. Math., 6 (2004), 259-277. doi: 10.1142/S021919970400132X.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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