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Weak closed-loop solvability of stochastic linear-quadratic optimal control problems
Existence and stability of one-peak symmetric stationary solutions for the Schnakenberg model with 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 $ |
$ g(x) $ |
$ g(x) = g(-x) $ |
$ \varepsilon>0 $ |
$ g(x) $ |
$ g(x) = 1 $ |
$ d = 0 $ |
$ N- $ |
$ (u_{\varepsilon}, v_{\varepsilon}) $ |
$ (u_{\varepsilon}, v_{\varepsilon}) $ |
$ g(x) = 1 $ |
References:
[1] |
S. Agmon, Lectures on Exponential Decay of Solutions of Second-Orderer Elliptic Equations, Princeton Univ. Press, 1982.
![]() ![]() |
[2] |
D. L. Benson, J. A. Sherrat and P. K. Maini,
Diffusion driven instability in an inhomogeneous domain, Bull. of Math. Biology, 55 (1993), 365-384.
|
[3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[4] |
K. Ikeda,
The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system, Netw. Heterog. Media, 8 (2013), 291-325.
doi: 10.3934/nhm.2013.8.291. |
[5] |
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. |
[6] |
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. |
[7] |
E. H. Lieb and M. Loss, Analysis, Vol.14 of Graduate Studies in Mathematics, American Math. Society, 2001.
doi: 10.1090/gsm/014. |
[8] |
P. Liu, J. Shi, Y. Wang and X. Feng,
Bifurcation analysis of reaction-diffusion Scnakenberg model, J. Math. Chem., 51 (2013), 2001-2019.
doi: 10.1007/s10910-013-0196-x. |
[9] |
K. Morimoto,
Point-condensation phenomena and saturation effect for the one-dimensional Gierer-Meinhardt system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 973-995.
doi: 10.1016/j.anihpc.2010.01.003. |
[10] |
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. |
[11] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc., B237 (1952), 37-72.
|
[12] |
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. |
[13] |
J. Wei,
On single interior spike layer solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, Eur. J. Appl. Math., 10 (1999), 353-378.
doi: 10.1017/S0956792599003770. |
[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, Mathematical Aspects of Pattern Formation in Biological Systems, Vol. 189 of Applied Mathematical Sciences, Springer, London, 2014.
doi: 10.1007/978-1-4471-5526-3. |
[16] |
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] |
S. Agmon, Lectures on Exponential Decay of Solutions of Second-Orderer Elliptic Equations, Princeton Univ. Press, 1982.
![]() ![]() |
[2] |
D. L. Benson, J. A. Sherrat and P. K. Maini,
Diffusion driven instability in an inhomogeneous domain, Bull. of Math. Biology, 55 (1993), 365-384.
|
[3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[4] |
K. Ikeda,
The existence and uniqueness of unstable eigenvalues for stripe patterns in the Gierer-Meinhardt system, Netw. Heterog. Media, 8 (2013), 291-325.
doi: 10.3934/nhm.2013.8.291. |
[5] |
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. |
[6] |
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. |
[7] |
E. H. Lieb and M. Loss, Analysis, Vol.14 of Graduate Studies in Mathematics, American Math. Society, 2001.
doi: 10.1090/gsm/014. |
[8] |
P. Liu, J. Shi, Y. Wang and X. Feng,
Bifurcation analysis of reaction-diffusion Scnakenberg model, J. Math. Chem., 51 (2013), 2001-2019.
doi: 10.1007/s10910-013-0196-x. |
[9] |
K. Morimoto,
Point-condensation phenomena and saturation effect for the one-dimensional Gierer-Meinhardt system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 27 (2010), 973-995.
doi: 10.1016/j.anihpc.2010.01.003. |
[10] |
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. |
[11] |
A. M. Turing,
The chemical basis of morphogenesis, Philos. Trans. R. Soc., B237 (1952), 37-72.
|
[12] |
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. |
[13] |
J. Wei,
On single interior spike layer solutions of Gierer-Meinhardt system: uniqueness and spectrum estimates, Eur. J. Appl. Math., 10 (1999), 353-378.
doi: 10.1017/S0956792599003770. |
[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, Mathematical Aspects of Pattern Formation in Biological Systems, Vol. 189 of Applied Mathematical Sciences, Springer, London, 2014.
doi: 10.1007/978-1-4471-5526-3. |
[16] |
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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