    June  2020, 19(6): 2965-3031. doi: 10.3934/cpaa.2020130

## Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity

 Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami-Ohsawa, Hachioji, Tokyo 192-0397, JAPAN

Received  February 2019 Revised  October 2019 Published  March 2020

In this paper, we consider the following one-dimensional Schnakenberg model with periodic heterogeneity:
 $\begin{equation*} \begin{cases} u_t-\varepsilon ^2 u_{xx} = d\varepsilon -u+g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ \varepsilon v_t-Dv_{xx} = \frac{1}{2}-\frac{c}{\varepsilon}g(x)u^2 v , & x \in (-1,1) ,\; t>0, \\ u_x (\pm 1) = v_x (\pm 1) = 0 .\end{cases} \end{equation*}$
where
 $d,c,D>0$
are given constants,
 $\varepsilon >0$
is sufficiently small, and
 $g(x)$
is a given positive function. Let
 $N \ge 1$
be an arbitrary natural number. We assume that
 $g(x)$
is a periodic and symmetric function, namely
 $g(x) = g(-x)$
and
 $g(x) = g(x+2N^{-1})$
. We study the stability of
 $N$
-peak stationary symmetric solutions. In particular, we are interested in the effect of the periodic heterogeneity
 $g(x)$
above on their stability. For the standard Schnakenberg model, namely the case of
 $g(x) = 1$
, with
 $d = 0$
, the stability of
 $N$
-peak solutions was established by Iron, Wei, and Winter in 2004. In this paper, we rigorously give a linear stability analysis and reveal the effect of the periodic heterogeneity on the stability of
 $N$
-peak solution. In particular, we investigate how
 $N$
-peak solutions is stabilized or destabilized by the effect of periodic heterogeneity compared with the case
 $g(x) = 1$
.
Citation: Yuta Ishii. Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity. Communications on Pure & Applied Analysis, 2020, 19 (6) : 2965-3031. doi: 10.3934/cpaa.2020130
##### References:
  W. Ao and C. Liu, The Schnakenberg model with precursors, Discrete Contin. Dyn. Syst., 39 (2019), 1923-1955.  doi: 10.3934/dcds.2019081.  Google Scholar  H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar  A. Doelman, A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: a matched asymptotic approach, Phys. D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar  A. Doelman, A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., 155 (2002), xii+64. doi: 10.1090/memo/0737.  Google Scholar  A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563.  doi: 10.1088/0951-7715/10/2/013.  Google Scholar  D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390.  doi: 10.1007/s00285-003-0258-y.  Google Scholar  Y. Ishii and K. Kurata, Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity, Discrete Contin. Dyn. Syst., 39 (2019), 2807-2875.  doi: 10.3934/dcds.2019118.  Google Scholar  T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71.  doi: 10.1111/j.1467-9590.2005.01554.  Google Scholar  T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.  Google Scholar  T. Kolokolnikov and J. Wei, Pattern formation in a reaction-diffusion system with space-dependent feed rate, SIAM Rev., 60 (2018), 626-645.  doi: 10.1137/17M1116027.  Google Scholar  T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Physica D, 2019. Available from: https://doi.org/10.1016/j.physd.2019.132247. doi: 10.1016/j.physd.2019.132247.  Google Scholar  J. Schnakenberg, Simple chemical reaction system with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.  Google Scholar  M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264.  doi: 10.1111/1467-9590.00223.  Google Scholar  J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion system, J. Math. Biol., 57 (2008), 53-89.  doi: 10.1007/s00285-007-0146-y.  Google Scholar  J. Wei and M. Winter, On the Gierer-Meinhardt system with precursors, Discrete Contin. Dyn. Syst., 25 (2009), 363-398.  doi: 10.3934/dcds.2009.25.363.  Google Scholar  J. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetic, J. Math. Biol., 64 (2012), 211-254.  doi: 10.1007/s00285-011-0412-x.  Google Scholar  J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, Vol. 189, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.  Google Scholar  J. Wei and M. Winter, Stable spike clusters for the one-dimensional Gierer-Meinhardt system, Eur. J. Appl. Math., 28 (2017), 576-635.  doi: 10.1017/S0956792516000450.  Google Scholar

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##### References:
  W. Ao and C. Liu, The Schnakenberg model with precursors, Discrete Contin. Dyn. Syst., 39 (2019), 1923-1955.  doi: 10.3934/dcds.2019081.  Google Scholar  H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. Google Scholar  A. Doelman, A. Gardner and T. J. Kaper, Stability analysis of singular patterns in the 1-D Gray-Scott model: a matched asymptotic approach, Phys. D, 122 (1998), 1-36.  doi: 10.1016/S0167-2789(98)00180-8.  Google Scholar  A. Doelman, A. Gardner and T. J. Kaper, A stability index analysis of 1-D patterns of the Gray-Scott model, Mem. Amer. Math. Soc., 155 (2002), xii+64. doi: 10.1090/memo/0737.  Google Scholar  A. Doelman, T. J. Kaper and P. A. Zegeling, Pattern formation in the one-dimensional Gray-Scott model, Nonlinearity, 10 (1997), 523-563.  doi: 10.1088/0951-7715/10/2/013.  Google Scholar  D. Iron, J. Wei and M. Winter, Stability analysis of Turing patterns generated by the Schnakenberg model, J. Math. Biol., 49 (2004), 358-390.  doi: 10.1007/s00285-003-0258-y.  Google Scholar  Y. Ishii and K. Kurata, Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with heterogeneity, Discrete Contin. Dyn. Syst., 39 (2019), 2807-2875.  doi: 10.3934/dcds.2019118.  Google Scholar  T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the low feed-rate regime, Stud. Appl. Math., 115 (2005), 21-71.  doi: 10.1111/j.1467-9590.2005.01554.  Google Scholar  T. Kolokolnikov, M. J. Ward and J. Wei, The existence and stability of spike equilibria in the one-dimensional Gray-Scott model: the pulse-splitting regime, Phys. D, 202 (2005), 258-293.  doi: 10.1016/j.physd.2005.02.009.  Google Scholar  T. Kolokolnikov and J. Wei, Pattern formation in a reaction-diffusion system with space-dependent feed rate, SIAM Rev., 60 (2018), 626-645.  doi: 10.1137/17M1116027.  Google Scholar  T. Kolokolnikov and S. Xie, Spike density distribution for the Gierer-Meinhardt model with precursor, Physica D, 2019. Available from: https://doi.org/10.1016/j.physd.2019.132247. doi: 10.1016/j.physd.2019.132247.  Google Scholar  J. Schnakenberg, Simple chemical reaction system with limit cycle behaviour, J. Theor. Biol., 81 (1979), 389-400.  doi: 10.1016/0022-5193(79)90042-0.  Google Scholar  M. J. Ward and J. Wei, The existence and stability of asymmetric spike patterns for the Schnakenberg model, Stud. Appl. Math., 109 (2002), 229-264.  doi: 10.1111/1467-9590.00223.  Google Scholar  J. Wei and M. Winter, Stationary multiple spots for reaction-diffusion system, J. Math. Biol., 57 (2008), 53-89.  doi: 10.1007/s00285-007-0146-y.  Google Scholar  J. Wei and M. Winter, On the Gierer-Meinhardt system with precursors, Discrete Contin. Dyn. Syst., 25 (2009), 363-398.  doi: 10.3934/dcds.2009.25.363.  Google Scholar  J. Wei and M. Winter, Flow-distributed spikes for Schnakenberg kinetic, J. Math. Biol., 64 (2012), 211-254.  doi: 10.1007/s00285-011-0412-x.  Google Scholar  J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Applied Mathematical Sciences, Vol. 189, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3.  Google Scholar  J. Wei and M. Winter, Stable spike clusters for the one-dimensional Gierer-Meinhardt system, Eur. J. Appl. Math., 28 (2017), 576-635.  doi: 10.1017/S0956792516000450.  Google Scholar For $D_2^{2,+}(\xi_2) > D = (3+\sqrt{17})/16-0.1$, two-peak solution is stable. For $D_2^{2,+}(\xi_2) < D = (3+\sqrt{17})/16+0.1$, two-peak solution is unstable
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