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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

 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

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.
Citation: Shin-Ichiro Ei, Kota Ikeda, Eiji Yanagida. Instability of multi-spot patterns in shadow systems of reaction-diffusion equations. Communications on Pure & 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,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [2] H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$,, \emph{Indiana Univ. Math. J.}, 30 (1981), 141. Google Scholar [3] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system,, \emph{Adv. Differential Equations}, 6 (2001), 847. doi: 10.1137/S0036141099364954. Google Scholar [4] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, \emph{Indiana Univ. Math. J.}, 50 (2001), 443. Google Scholar [5] S.-I. Ei, K. Ikeda and Y. Miyamoto, Dynamics of a boundary spike for the shadow Gierer-Meinhardt system,, \emph{Commun. Pure Appl. Anal.}, 11 (2012), 115. Google Scholar [6] L. C. Evans, Partial Differential Equations,, vol.19 of Graduate Studies in Mathematics, (1998). Google Scholar [7] C. Gui, Multipeak solutions for a semilinear Neumann problem,, \emph{Duke Math. J.}, 84 (1996), 739. doi: 10.1215/S0012-7094-96-08423-9. Google Scholar [8] K. Ikeda, The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system,, \emph{Netw. Heterog. Media}, 8 (2013), 291. doi: 10.3934/nhm.2013.8.291. Google Scholar [9] T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995). Google Scholar [10] K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 1443. doi: 10.3934/cpaa.2008.7.1443. Google Scholar [11] C. S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem,, in \emph{Calculus of variations and partial differential equations (Trento, (1988), 160. doi: 10.1007/BFb0082894. Google Scholar [12] Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball,, \emph{J. Differential Equations}, 229 (2006), 494. doi: 10.1016/j.jde.2006.03.015. Google Scholar [13] W.-M. Ni, P. PolJáčik and E. Yanagida, Monotonicity of stable solutions in shadow systems,, \emph{Trans. Amer. Math. Soc.}, 353 (2001), 5057. Google Scholar [14] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, \emph{Duke Math. J.}, 70 (1993), 247. doi: 10.1215/S0012-7094-93-07004-4. Google Scholar [15] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems,, \emph{SIAM J. Math. Anal.}, 13 (1982), 555. doi: 10.1137/0513037. Google Scholar [16] I. Takagi, Point-condensation for a reaction-diffusion system,, \emph{J. Differential Equations}, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. Google Scholar [17] T. Wakasa and S. Yotsutani, Representation formulas for some 1-dimensional linearized eigenvalue problems,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 745. doi: 10.3934/cpaa.2008.7.745. Google Scholar [18] J. Wei, Existence, stability and metastability of point condensation patterns generated by the Gray-Scott system,, \emph{Nonlinearity}, 12 (1999), 593. doi: 10.1088/0951-7715/12/3/011. Google Scholar [19] J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $R^2$,, \emph{Phys. D}, 176 (2003), 147. doi: 10.1016/S0167-2789(02)00743-1. Google Scholar [20] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, \emph{J. Math. Pures Appl.}, 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. Google Scholar [21] J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $R^1$,, \emph{Methods Appl. Anal.}, 14 (2007), 119. doi: 10.4310/MAA.2007.v14.n2.a2. Google Scholar

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##### References:
 [1] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state,, \emph{Arch. Rational Mech. Anal.}, 82 (1983), 313. doi: 10.1007/BF00250555. Google Scholar [2] H. Berestycki, P.-L. Lions and L. A. Peletier, An ODE approach to the existence of positive solutions for semilinear problems in $R^N$,, \emph{Indiana Univ. Math. J.}, 30 (1981), 141. Google Scholar [3] X. Chen and M. Kowalczyk, Slow dynamics of interior spikes in the shadow Gierer-Meinhardt system,, \emph{Adv. Differential Equations}, 6 (2001), 847. doi: 10.1137/S0036141099364954. Google Scholar [4] A. Doelman, R. Gardner and T. J. Kaper, Large stable pulse solutions in reaction-diffusion equations,, \emph{Indiana Univ. Math. J.}, 50 (2001), 443. Google Scholar [5] S.-I. Ei, K. Ikeda and Y. Miyamoto, Dynamics of a boundary spike for the shadow Gierer-Meinhardt system,, \emph{Commun. Pure Appl. Anal.}, 11 (2012), 115. Google Scholar [6] L. C. Evans, Partial Differential Equations,, vol.19 of Graduate Studies in Mathematics, (1998). Google Scholar [7] C. Gui, Multipeak solutions for a semilinear Neumann problem,, \emph{Duke Math. J.}, 84 (1996), 739. doi: 10.1215/S0012-7094-96-08423-9. Google Scholar [8] K. Ikeda, The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system,, \emph{Netw. Heterog. Media}, 8 (2013), 291. doi: 10.3934/nhm.2013.8.291. Google Scholar [9] T. Kato, Perturbation Theory for Linear Operators,, Classics in Mathematics, (1995). Google Scholar [10] K. Kurata and K. Morimoto, Construction and asymptotic behavior of multi-peak solutions to the Gierer-Meinhardt system with saturation,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 1443. doi: 10.3934/cpaa.2008.7.1443. Google Scholar [11] C. S. Lin and W.-M. Ni, On the diffusion coefficient of a semilinear Neumann problem,, in \emph{Calculus of variations and partial differential equations (Trento, (1988), 160. doi: 10.1007/BFb0082894. Google Scholar [12] Y. Miyamoto, An instability criterion for activator-inhibitor systems in a two-dimensional ball,, \emph{J. Differential Equations}, 229 (2006), 494. doi: 10.1016/j.jde.2006.03.015. Google Scholar [13] W.-M. Ni, P. PolJáčik and E. Yanagida, Monotonicity of stable solutions in shadow systems,, \emph{Trans. Amer. Math. Soc.}, 353 (2001), 5057. Google Scholar [14] W.-M. Ni and I. Takagi, Locating the peaks of least-energy solutions to a semilinear Neumann problem,, \emph{Duke Math. J.}, 70 (1993), 247. doi: 10.1215/S0012-7094-93-07004-4. Google Scholar [15] Y. Nishiura, Global structure of bifurcating solutions of some reaction-diffusion systems,, \emph{SIAM J. Math. Anal.}, 13 (1982), 555. doi: 10.1137/0513037. Google Scholar [16] I. Takagi, Point-condensation for a reaction-diffusion system,, \emph{J. Differential Equations}, 61 (1986), 208. doi: 10.1016/0022-0396(86)90119-1. Google Scholar [17] T. Wakasa and S. Yotsutani, Representation formulas for some 1-dimensional linearized eigenvalue problems,, \emph{Commun. Pure Appl. Anal.}, 7 (2008), 745. doi: 10.3934/cpaa.2008.7.745. Google Scholar [18] J. Wei, Existence, stability and metastability of point condensation patterns generated by the Gray-Scott system,, \emph{Nonlinearity}, 12 (1999), 593. doi: 10.1088/0951-7715/12/3/011. Google Scholar [19] J. Wei and M. Winter, Existence and stability of multiple-spot solutions for the Gray-Scott model in $R^2$,, \emph{Phys. D}, 176 (2003), 147. doi: 10.1016/S0167-2789(02)00743-1. Google Scholar [20] J. Wei and M. Winter, Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system,, \emph{J. Math. Pures Appl.}, 83 (2004), 433. doi: 10.1016/j.matpur.2003.09.006. Google Scholar [21] J. Wei and M. Winter, Existence, classification and stability analysis of multiple-peaked solutions for the Gierer-Meinhardt system in $R^1$,, \emph{Methods Appl. Anal.}, 14 (2007), 119. doi: 10.4310/MAA.2007.v14.n2.a2. Google Scholar
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