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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  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 & Applied Analysis, 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] Y. Ishii, The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model, submitted. Google Scholar [12] Y. Ishii, Stability analysis of spike solutions to the Schnakenberg model with heterogeneity on metric graphs, submitted. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [4] H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [8] A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.  doi: 10.1007/BF00289234.  Google Scholar [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.  Google Scholar [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.  Google Scholar [11] Y. Ishii, The effect of heterogeneity on one-peak stationary solutions to the Schnakenberg model, submitted. Google Scholar [12] Y. Ishii, Stability analysis of spike solutions to the Schnakenberg model with heterogeneity on metric graphs, submitted. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
$Y$-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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