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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 |
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 [
Mathematics Subject Classification:
Primary 35B25, 35R02; Secondary 35K57, 35Q92.
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
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References:
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W. Ao and C. Liu,
The Schnakenberg model with precursors, Discrete Contin. Dyn. Syst., 39 (2019), 1923-1955.
doi: 10.3934/dcds.2019081. |
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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. |
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H. Brezis, Functional analysis, Sobolev spaces and partial differential equations, Universitext, Springer, New York, 2011. |
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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. |
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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. |
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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. |
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A. Gierer and H. Meinhardt,
A theory of biological pattern formation, Kybernetik, 12 (1972), 30-39.
doi: 10.1007/BF00289234. |
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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. |
show all references
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. |
Figure 2.
A one-peak solution and a two-peak solution on the interval $ (-1, 1) $ . $ L $ is a length of $ (-1,1) $
Figure 3.
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 $
Figure 4.
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
Figure 5.
$ H $ -shaped metric graph
Figure 6.
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 $
Figure 7.
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
Figure 8.
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) $
Figure 9.
$ \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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