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Classification of singular sets of solutions to elliptic equations
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 |
$ \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*} $ |
$ d,c,D>0 $ |
$ \varepsilon >0 $ |
$ g(x) $ |
$ N \ge 1 $ |
$ g(x) $ |
$ g(x) = g(-x) $ |
$ g(x) = g(x+2N^{-1}) $ |
$ N $ |
$ g(x) $ |
$ g(x) = 1 $ |
$ d = 0 $ |
$ N $ |
$ N $ |
$ N $ |
$ g(x) = 1 $ |
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] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[3] |
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. |
[4] |
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. |
[5] |
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. |
[6] |
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. |
[7] |
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. |
[8] |
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. |
[9] |
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. |
[10] |
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. |
[11] |
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. |
[12] |
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. |
[13] |
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. |
[14] |
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. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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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