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Asymptotic expansion of the mean-field approximation
The Schnakenberg model with precursors
School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China |
$ \mu(x) $ |
$ (-1,1) $ |
$ \begin{equation*}\left\{\begin{array}{l}u_{t} = D_{1}u''-\mu(x)u+vu^{2} \hspace{1.64cm} \text{in }(-1,1),\\v_{t} = D_{2}v''+B-vu^{2}\hspace{2.2cm} \;\;\;\; \text{in } (-1,1),\\u'(\pm1) = v'(\pm1) = 0,\end{array}\right.\end{equation*}$ |
$ D_{1}>0 $ |
$ D_{2}>0 $ |
$ B>0 $ |
$ N- $ |
$ \mu(x) $ |
$ D_{1} $ |
$ D_{2} $ |
$ \mu(x) $ |
References:
[1] |
W. W. Ao, M. Musso and J. C. Wei,
On spikes concentrating on line-segments to a semilinear Neumann problem, Journal of Differential Equations, 251 (2011), 881-901.
doi: 10.1016/j.jde.2011.05.009. |
[2] |
D. Benson, P. Maini and J. Sherratt,
Unravelling the Turing bifurcation using spatially varying diffusion cofficents, J. Math.Biol, 37 (1998), 381-417.
doi: 10.1007/s002850050135. |
[3] |
A. Floer and A. Weinstin,
Nonspreading wave packets for the cubic Schödinger equations with a bounded potential, J.Functional Anailsis, 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[4] |
D. Gilbarg and N. Turdinger, Elliptic Partical Differential Equations of Second Order, Springer, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[5] |
C. F. Gui and J. C. Wei,
Multiple interior peak solutions for some singularly perturbation problems, J. Differential Equations, 158 (1999), 1-27.
doi: 10.1016/S0022-0396(99)80016-3. |
[6] |
C. F. Gui, J. C. Wei and M. Winter,
Multiple boundary peak solutions for some singularly peturbed Neumman problems, Ann. Inst. H. Poincaré Anal., 17 (2000), 47-82.
doi: 10.1016/S0294-1449(99)00104-3. |
[7] |
D. Iron, J. C. 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. |
[8] |
Y. G. Oh.,
Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V)α, Comm.partia Differential Equations, 13 (1990), 1499-1519.
doi: 10.1080/03605308808820585. |
[9] |
Y. G. Oh,
On positive multi-bump bound states of nonlinear Schrödinger equations under multiple-well potentials, Comm.Math.Phys., 131 (1990), 223-253.
doi: 10.1007/BF02161413. |
[10] |
J. Schnakenberg,
Simple chemical reaction systems with limit cycle behavior, J.Theoret.Biol, 81 (1979), 389-400.
doi: 10.1016/0022-5193(79)90042-0. |
[11] |
A. Turing,
The chemical basis of morphogenesis, Phil. Trans.Roy, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[12] |
M. Ward and J. C. Wei,
Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability, J.Appl.Math, 13 (2002), 283-320.
doi: 10.1017/S0956792501004442. |
[13] |
M. Ward and J. C. Wei,
The existence and stability of asymmetric spike partterns for the Schnakenberg model, Michigan Math. J., 109 (2002), 229-264.
doi: 10.1111/1467-9590.00223. |
[14] |
J. C. Wei,
On single interior spike solutions of Gierer-Meinhardt system: Uniqueness and spectrum estimates, European. J. Appl. Mth., 10 (1999), 353-378.
doi: 10.1017/S0956792599003770. |
[15] |
J. C. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst.H.Poincaré Anal., 348 (1996), 975-995. Google Scholar |
[16] |
J. C. Wei and M. Winter,
On the Cahn-Hilliard equations: Interior spike layer solutions, J. Differential Equations, 148 (1998), 231-267.
doi: 10.1006/jdeq.1998.3479. |
[17] |
J. C. Wei and M. Winter,
Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case, J.Nonlinear Science, 11 (2001), 415-458.
doi: 10.1007/s00332-001-0380-1. |
[18] |
J. C. Wei and M. Winter,
Spikes for the Gierer-Meinhardt system in the two dimensions: The weak coupling case, J.Differential Equations, 178 (2002), 478-518.
doi: 10.1006/jdeq.2001.4019. |
[19] |
J. C. Wei and M. Winter,
Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math.Pures.Appl, 83 (2004), 433-476.
doi: 10.1016/j.matpur.2003.09.006. |
[20] |
J. C. Wei and M. Winter,
Existence, Classification and Stability analysis of multiple-peaked soiltion for the Gierer-Meinhardt system in R1, Methods and Applications of Analysis, 14 (2007), 119-163.
doi: 10.4310/MAA.2007.v14.n2.a2. |
[21] |
J. C. 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. |
[22] |
J. C. Wei and M. Winter,
Flow-distributed spikes for Schnakenberg Kinetics, Mathematical Biology, 64 (2012), 211-254.
doi: 10.1007/s00285-011-0412-x. |
show all references
References:
[1] |
W. W. Ao, M. Musso and J. C. Wei,
On spikes concentrating on line-segments to a semilinear Neumann problem, Journal of Differential Equations, 251 (2011), 881-901.
doi: 10.1016/j.jde.2011.05.009. |
[2] |
D. Benson, P. Maini and J. Sherratt,
Unravelling the Turing bifurcation using spatially varying diffusion cofficents, J. Math.Biol, 37 (1998), 381-417.
doi: 10.1007/s002850050135. |
[3] |
A. Floer and A. Weinstin,
Nonspreading wave packets for the cubic Schödinger equations with a bounded potential, J.Functional Anailsis, 69 (1986), 397-408.
doi: 10.1016/0022-1236(86)90096-0. |
[4] |
D. Gilbarg and N. Turdinger, Elliptic Partical Differential Equations of Second Order, Springer, Berlin, 1983.
doi: 10.1007/978-3-642-61798-0. |
[5] |
C. F. Gui and J. C. Wei,
Multiple interior peak solutions for some singularly perturbation problems, J. Differential Equations, 158 (1999), 1-27.
doi: 10.1016/S0022-0396(99)80016-3. |
[6] |
C. F. Gui, J. C. Wei and M. Winter,
Multiple boundary peak solutions for some singularly peturbed Neumman problems, Ann. Inst. H. Poincaré Anal., 17 (2000), 47-82.
doi: 10.1016/S0294-1449(99)00104-3. |
[7] |
D. Iron, J. C. 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. |
[8] |
Y. G. Oh.,
Existence of semi-classical bound states of nonlinear Schrödinger equations with potentials of the class (V)α, Comm.partia Differential Equations, 13 (1990), 1499-1519.
doi: 10.1080/03605308808820585. |
[9] |
Y. G. Oh,
On positive multi-bump bound states of nonlinear Schrödinger equations under multiple-well potentials, Comm.Math.Phys., 131 (1990), 223-253.
doi: 10.1007/BF02161413. |
[10] |
J. Schnakenberg,
Simple chemical reaction systems with limit cycle behavior, J.Theoret.Biol, 81 (1979), 389-400.
doi: 10.1016/0022-5193(79)90042-0. |
[11] |
A. Turing,
The chemical basis of morphogenesis, Phil. Trans.Roy, 237 (1952), 37-72.
doi: 10.1098/rstb.1952.0012. |
[12] |
M. Ward and J. C. Wei,
Asymmetric spike patterns for the one-dimensional Gierer-Meinhardt model: equilibria and stability, J.Appl.Math, 13 (2002), 283-320.
doi: 10.1017/S0956792501004442. |
[13] |
M. Ward and J. C. Wei,
The existence and stability of asymmetric spike partterns for the Schnakenberg model, Michigan Math. J., 109 (2002), 229-264.
doi: 10.1111/1467-9590.00223. |
[14] |
J. C. Wei,
On single interior spike solutions of Gierer-Meinhardt system: Uniqueness and spectrum estimates, European. J. Appl. Mth., 10 (1999), 353-378.
doi: 10.1017/S0956792599003770. |
[15] |
J. C. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst.H.Poincaré Anal., 348 (1996), 975-995. Google Scholar |
[16] |
J. C. Wei and M. Winter,
On the Cahn-Hilliard equations: Interior spike layer solutions, J. Differential Equations, 148 (1998), 231-267.
doi: 10.1006/jdeq.1998.3479. |
[17] |
J. C. Wei and M. Winter,
Spikes for the two-dimensional Gierer-Meinhardt system: The weak coupling case, J.Nonlinear Science, 11 (2001), 415-458.
doi: 10.1007/s00332-001-0380-1. |
[18] |
J. C. Wei and M. Winter,
Spikes for the Gierer-Meinhardt system in the two dimensions: The weak coupling case, J.Differential Equations, 178 (2002), 478-518.
doi: 10.1006/jdeq.2001.4019. |
[19] |
J. C. Wei and M. Winter,
Existence and stability analysis of asymmetric patterns for the Gierer-Meinhardt system, J. Math.Pures.Appl, 83 (2004), 433-476.
doi: 10.1016/j.matpur.2003.09.006. |
[20] |
J. C. Wei and M. Winter,
Existence, Classification and Stability analysis of multiple-peaked soiltion for the Gierer-Meinhardt system in R1, Methods and Applications of Analysis, 14 (2007), 119-163.
doi: 10.4310/MAA.2007.v14.n2.a2. |
[21] |
J. C. 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. |
[22] |
J. C. Wei and M. Winter,
Flow-distributed spikes for Schnakenberg Kinetics, Mathematical Biology, 64 (2012), 211-254.
doi: 10.1007/s00285-011-0412-x. |
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