Instability of multi-spot patterns in shadow systems of reaction-diffusion equations

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March 2015, 14(2): 717-736. doi: 10.3934/cpaa.2015.14.717

Instability of multi-spot patterns in shadow systems of reaction-diffusion equations

Shin-Ichiro Ei ^1, , Kota Ikeda ^2, and Eiji Yanagida ^3,

1.	Department of Mathematics, Faculty of Science, Hokkaido University, Sapporo, 060-0810, Japan
2.	Department of Mathematical Sciences Based on Modeling and Analysis, Meiji University, Nakano-ku, Tokyo, 164-8525, Japan
3.	Department of Mathematics, Tokyo Institute of Technology, Meguro-ku, Tokyo 152-8551

Received July 2013 Revised December 2013 Published December 2014

Abstract
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Our aim in this paper is to prove the instability of multi-spot patterns in a shadow system, which is obtained as a limiting system of a reaction-diffusion model as one of the diffusion coefficients goes to infinity. Instead of investigating each eigenfunction for a linearized operator, we characterize the eigenspace spanned by unstable eigenfunctions.

Keywords: reaction-diffusion equation, Eigenpair, shadow system, stability, multi-spot solution.

Mathematics Subject Classification: Primary: 35B36, 35K57; Secondary: 37L15.

Citation: Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 717-736. doi: 10.3934/cpaa.2015.14.717

References:

[1]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[2]	H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$, Indiana Univ. Math. J., 30 (1981), 141-157.
[3]	X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system, Adv. Differential Equations, 6 (2001), 847-872. doi: 10.1137/S0036141099364954.
[4]	A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507.
[5]	S.-I. Ei, K. Ikeda and Y. Miyamoto, Dynamics of a boundary spike for the shadow Gierer-Meinhardt system, Commun. Pure Appl. Anal., 11 (2012), 115-145.
[6]	L. C. Evans, Partial Differential Equations, vol.19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.
[7]	C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769. doi: 10.1215/S0012-7094-96-08423-9.
[8]	K. Ikeda, The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system, Netw. Heterog. Media, 8 (2013), 291-325. doi: 10.3934/nhm.2013.8.291.
[9]	T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
[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]	C. S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem, in Calculus of variations and partial differential equations (Trento, 1986), vol.1340 of Lecture Notes in Math., Springer, Berlin, (1988), 160-174. doi: 10.1007/BFb0082894.
[12]	Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball, J. Differential Equations, 229 (2006), 494-508. doi: 10.1016/j.jde.2006.03.015.
[13]	W.-M. Ni, P. PolJáčik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. Amer. Math. Soc., 353 (2001), 5057-5069 (electronic).
[14]	W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[15]	Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037.
[16]	I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249. doi: 10.1016/0022-0396(86)90119-1.
[17]	T. Wakasa and S. Yotsutani, Representation formulas for some 1-dimensional linearized eigenvalue problems, Commun. Pure Appl. Anal., 7 (2008), 745-763. doi: 10.3934/cpaa.2008.7.745.
[18]	J. Wei, Existence, stability and metastability of point condensation patterns generated by the Gray-Scott system, Nonlinearity, 12 (1999), 593-616. doi: 10.1088/0951-7715/12/3/011.
[19]	J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $R^2$, Phys. D, 176 (2003), 147-180. doi: 10.1016/S0167-2789(02)00743-1.
[20]	J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl., 83 (2004), 433-476. doi: 10.1016/j.matpur.2003.09.006.
[21]	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. doi: 10.4310/MAA.2007.v14.n2.a2.

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References:

[1]	H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal., 82 (1983), 313-345. doi: 10.1007/BF00250555.
[2]	H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$, Indiana Univ. Math. J., 30 (1981), 141-157.
[3]	X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system, Adv. Differential Equations, 6 (2001), 847-872. doi: 10.1137/S0036141099364954.
[4]	A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations, Indiana Univ. Math. J., 50 (2001), 443-507.
[5]	S.-I. Ei, K. Ikeda and Y. Miyamoto, Dynamics of a boundary spike for the shadow Gierer-Meinhardt system, Commun. Pure Appl. Anal., 11 (2012), 115-145.
[6]	L. C. Evans, Partial Differential Equations, vol.19 of Graduate Studies in Mathematics, American Mathematical Society, Providence, RI, 1998.
[7]	C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J., 84 (1996), 739-769. doi: 10.1215/S0012-7094-96-08423-9.
[8]	K. Ikeda, The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system, Netw. Heterog. Media, 8 (2013), 291-325. doi: 10.3934/nhm.2013.8.291.
[9]	T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition.
[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]	C. S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem, in Calculus of variations and partial differential equations (Trento, 1986), vol.1340 of Lecture Notes in Math., Springer, Berlin, (1988), 160-174. doi: 10.1007/BFb0082894.
[12]	Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball, J. Differential Equations, 229 (2006), 494-508. doi: 10.1016/j.jde.2006.03.015.
[13]	W.-M. Ni, P. PolJáčik and E. Yanagida, Monotonicity of stable solutions in shadow systems, Trans. Amer. Math. Soc., 353 (2001), 5057-5069 (electronic).
[14]	W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J., 70 (1993), 247-281. doi: 10.1215/S0012-7094-93-07004-4.
[15]	Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems, SIAM J. Math. Anal., 13 (1982), 555-593. doi: 10.1137/0513037.
[16]	I. Takagi, Point-condensation for a reaction-diffusion system, J. Differential Equations, 61 (1986), 208-249. doi: 10.1016/0022-0396(86)90119-1.
[17]	T. Wakasa and S. Yotsutani, Representation formulas for some 1-dimensional linearized eigenvalue problems, Commun. Pure Appl. Anal., 7 (2008), 745-763. doi: 10.3934/cpaa.2008.7.745.
[18]	J. Wei, Existence, stability and metastability of point condensation patterns generated by the Gray-Scott system, Nonlinearity, 12 (1999), 593-616. doi: 10.1088/0951-7715/12/3/011.
[19]	J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $R^2$, Phys. D, 176 (2003), 147-180. doi: 10.1016/S0167-2789(02)00743-1.
[20]	J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math. Pures Appl., 83 (2004), 433-476. doi: 10.1016/j.matpur.2003.09.006.
[21]	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. doi: 10.4310/MAA.2007.v14.n2.a2.

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