# American Institute of Mathematical Sciences

April  2021, 20(4): 1633-1679. doi: 10.3934/cpaa.2021035

## Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs

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

* Corresponding author

Received  October 2020 Revised  January 2021 Published  April 2021 Early access  March 2021

Fund Project: The first author was supported by JSPS KAKENHI Grant Number 20J12212. The second author was supported by JSPS KAKENHI Grant Numbers 17H01092, 19K03587

In this paper, we study the existence of spiky stationary solutions of the Schnakenberg model with heterogeneity on compact metric graphs. These solutions are constructed by using the Liapunov–Schmidt reduction method and taking the same strategy as that in [14,11]. First, we give the abstract theorem on the existence of multi-peak solutions for general compact metric graphs under several assumptions for the associated Green's function. In particular, we reveal that how locations of concentration points and amplitudes of spiky solutions are determined by the interaction of the heterogeneity with the geometry of the compact metric graph, represented by Green's function. Second, we apply our abstract theorem to the $Y$-shaped metric graph and the $H$-shaped metric graph in non-heterogeneity case. In particular, we show the precise effect of the geometry of those compact graphs to the locations of concentration points for these concrete graphs, respectively.

Citation: Yuta Ishii, Kazuhiro Kurata. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Communications on Pure and Applied Analysis, 2021, 20 (4) : 1633-1679. doi: 10.3934/cpaa.2021035
##### References:
 [1] W. Ao and C. Liu, The Schnakenberg model with precursors, Discrete Contin. Dyn. Syst., 39 (2019), 1923-1955.  doi: 10.3934/dcds.2019081. [2] J. V. Below and J. A. Lubary, Instability of Stationary Solutions of Reaction-Diffusion-Equations on Graphs, Results. Math., 68 (2015), 171-201.  doi: 10.1007/s00025-014-0429-8. [3] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, in Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2013. doi: 10.1090/surv/186. [4] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. [5] F. Camilli and L. Corrias, Parabolic models for chemotaxis on weighted networks, J. Math. Pures Appl., 108 (2017), 459-480.  doi: 10.1016/j.matpur.2017.07.003. [6] Y. Du, B. Lou, R. Peng and M. Zhou, The Fisher-KPP equation over simple graphs: Varied persistence states in river networks, J. Math. Biol., 80 (2020), 1559-1616.  doi: 10.1007/s00285-020-01474-1. [7] P. Exner and H. Kova${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over r} }}}$ík, Quantum Waveguides, Theoretical and Mathematical Physics, Springer, Chem, 2015. doi: 10.1007/978-3-319-18576-7. [8] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234. [9] 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. [10] Y. Ishii, Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity, Commun. Pure Appl. Anal., 19(6) (2020) 2965–3031. doi: 10.3934/cpaa.2020130. [11] Y. Ishii, The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model, submitted. [12] Y. Ishii, Stability analysis of spike solutions to the Schnakenberg model with heterogeneity on metric graphs, submitted. [13] Y. Ishii, Concentration phenomena on $Y$-shaped metric graph for the Gierer-Meinhardt model with heterogeneity, Nonlinear Anal., 205 (2021), 112220. doi: 10.1016/j.na.2020.112220. [14] Y. Ishii and K. Kurata, Existence and stability of one-peak symmetric stationary solutions for Schnakenberg model with heterogeneity, Discrete Contin. Dyn. Syst., 39 (2019), 2807-2875.  doi: 10.3934/dcds.2019118. [15] S. Jimbo and Y. Morita, Entire solutions to reaction-diffusion equations in multiple half-lines with a junction, J. Differ. Equ., 267 (2019), 1247-1276.  doi: 10.1016/j.jde.2019.02.008. [16] Y. Jin, R. Peng and J. Shi, Population dynamics in river networks, J. Nonlinear Sci., 29 (2019), 2501-2545.  doi: 10.1007/s00332-019-09551-6. [17] S. Kosugi, A semilinear elliptic equation in a thin network-shaped domain, J. Math. Soc. Jpn., 52 (2000), 673-697.  doi: 10.2969/jmsj/05230673. [18] K. Kurata and M. Shibata, Least energy solutions to semi-linear elliptic problems on metric graphs, J. Math. Anal. Appl., 491 (2020), 124297. doi: 10.1016/j.jmaa.2020.124297. [19] Y. Li, F. Li and J. Shi, Ground states of nonlinear Schrödinger equation on star metric graphs, J. Math. Anal. Appl., 459 (2018), 661-685.  doi: 10.1016/j.jmaa.2017.10.069. [20] 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. [21] E. Yanagida, Stability of nonconstant steady states in reaction-diffusion systems on graphs, Japan J. Indust. Appl. Math., 18 (2001), 25-42.  doi: 10.1007/BF03167353. [22] 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. [23] 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. [24] J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3. [25] 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.

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##### References:
 [1] W. Ao and C. Liu, The Schnakenberg model with precursors, Discrete Contin. Dyn. Syst., 39 (2019), 1923-1955.  doi: 10.3934/dcds.2019081. [2] J. V. Below and J. A. Lubary, Instability of Stationary Solutions of Reaction-Diffusion-Equations on Graphs, Results. Math., 68 (2015), 171-201.  doi: 10.1007/s00025-014-0429-8. [3] G. Berkolaiko and P. Kuchment, Introduction to Quantum Graphs, in Mathematical Surveys and Monographs, American Mathematical Society, Providence, 2013. doi: 10.1090/surv/186. [4] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. [5] F. Camilli and L. Corrias, Parabolic models for chemotaxis on weighted networks, J. Math. Pures Appl., 108 (2017), 459-480.  doi: 10.1016/j.matpur.2017.07.003. [6] Y. Du, B. Lou, R. Peng and M. Zhou, The Fisher-KPP equation over simple graphs: Varied persistence states in river networks, J. Math. Biol., 80 (2020), 1559-1616.  doi: 10.1007/s00285-020-01474-1. [7] P. Exner and H. Kova${{\rm{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\smile$}} \over r} }}}$ík, Quantum Waveguides, Theoretical and Mathematical Physics, Springer, Chem, 2015. doi: 10.1007/978-3-319-18576-7. [8] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234. [9] 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. [10] Y. Ishii, Stability of multi-peak symmetric stationary solutions for the Schnakenberg model with periodic heterogeneity, Commun. Pure Appl. Anal., 19(6) (2020) 2965–3031. doi: 10.3934/cpaa.2020130. [11] Y. Ishii, The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model, submitted. [12] Y. Ishii, Stability analysis of spike solutions to the Schnakenberg model with heterogeneity on metric graphs, submitted. [13] Y. Ishii, Concentration phenomena on $Y$-shaped metric graph for the Gierer-Meinhardt model with heterogeneity, Nonlinear Anal., 205 (2021), 112220. doi: 10.1016/j.na.2020.112220. [14] Y. Ishii and K. Kurata, Existence and stability of one-peak symmetric stationary solutions for Schnakenberg model with heterogeneity, Discrete Contin. Dyn. Syst., 39 (2019), 2807-2875.  doi: 10.3934/dcds.2019118. [15] S. Jimbo and Y. Morita, Entire solutions to reaction-diffusion equations in multiple half-lines with a junction, J. Differ. Equ., 267 (2019), 1247-1276.  doi: 10.1016/j.jde.2019.02.008. [16] Y. Jin, R. Peng and J. Shi, Population dynamics in river networks, J. Nonlinear Sci., 29 (2019), 2501-2545.  doi: 10.1007/s00332-019-09551-6. [17] S. Kosugi, A semilinear elliptic equation in a thin network-shaped domain, J. Math. Soc. Jpn., 52 (2000), 673-697.  doi: 10.2969/jmsj/05230673. [18] K. Kurata and M. Shibata, Least energy solutions to semi-linear elliptic problems on metric graphs, J. Math. Anal. Appl., 491 (2020), 124297. doi: 10.1016/j.jmaa.2020.124297. [19] Y. Li, F. Li and J. Shi, Ground states of nonlinear Schrödinger equation on star metric graphs, J. Math. Anal. Appl., 459 (2018), 661-685.  doi: 10.1016/j.jmaa.2017.10.069. [20] 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. [21] E. Yanagida, Stability of nonconstant steady states in reaction-diffusion systems on graphs, Japan J. Indust. Appl. Math., 18 (2001), 25-42.  doi: 10.1007/BF03167353. [22] 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. [23] 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. [24] J. Wei and M. Winter, Mathematical Aspects of Pattern Formation in Biological Systems, Springer, London, 2014. doi: 10.1007/978-1-4471-5526-3. [25] 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.
$Y$-shaped metric graph
A one-peak solution and a two-peak solution on the interval $(-1, 1)$. $L$ is a length of $(-1,1)$
A concentration point $t_0$ of a one-peak solution on the $Y$-shaped graph. By Theorem 2.2, we have $A+l_2+l_3 = L/2$
Concentration points $t_1^0$, $t_2^0$ of a two-peak solution on the $Y$-shaped graph. Case A: By Theorem 2.3, it holds that $l_1 = l_2$ and $A_1+A_2+l_3 = L/2$. Moreover, we also have $A_1 = A_2$. Case B: A distance between $t_1^0$ and $t_2^0$ is $L/2$ and $A+l_2+l_3 = L/4$ holds
$H$-shaped metric graph
A concentration point of a one-peak solution on the $H$-shaped graph. If $t^0\in e_3$, by Theorem 3.2, then we have $A_1+l_1+l_2=L/2$ and $A_2+l_4+l_5=L/2$
Concentration points of a two-peak solution on the $H$-shaped graph (CaseA, Case C, and Case D). Case A: Using Theorem 3.3, we obtain $l_1 = l_2$. Case C: $A_1+A_2+l_2 = L/2$ and $l_1 = l_3+l_4+l_5$ is required. Then, we also have $A_1 = A_2$. Case D: A distance between $t_1^0$ and $t_2^0$ is $L/2$ and $A_1+l_1+l_2 = A_2+l_4+l_5 = L/4$ holds
Concentration points of a two-peak solution on the $H$-shaped graph (Case B). The point $B\in [0,l_3]$ divides $[0,l_3]$ internally in the ratio $l_5:l_2$. We have $A_1+B = A_2+(l_3-B)$
$\hat{\mathcal{G}}$ is an arbitrary metric and $\mathcal{G}$ is defined by $\mathcal{G}: = \hat{\mathcal{G}} \cup \{e\}$. If a edge $e$ is sufficiently long, then can we construct a one-peak solution which concentrates near $t^0 = l_e-L/2 \in e = [0,l_e]$?
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