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April  2019, 39(4): 1923-1955. doi: 10.3934/dcds.2019081

The Schnakenberg model with precursors

Weiwei Ao and  Chao Liu

School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China

Received  January 2018 Revised  March 2018 Published  January 2019

Fund Project: The research of the first author is supported by NSFC (No. 11801421 and No. 11631011).

In this paper, we mainly consider the following Schnakenberg model with a precursor
$ \mu(x) $
 on the interval
$ (-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*}$
where
$ D_{1}>0 $
,
$ D_{2}>0 $
,
$ B>0 $
.
We establish the existence and stability of
$ N- $
peaked steady-states in terms of the precursor
$ \mu(x) $
 and the diffusion coefficients
$ D_{1} $
 and
$ D_{2} $
. It is shown that
$ \mu(x) $
 plays an essential role for both existence and stability of the above pattern. Similar result has been obtained for the Gierer-Meinhardt system by Wei and Winter [21].
Keywords: Schnakenberg model, precursors, $ N- $peaked solutions, existence and stability.
Mathematics Subject Classification: Primary: 35J47; Secondary: 35J60.
Citation: Weiwei Ao, Chao Liu. The Schnakenberg model with precursors. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1923-1955. doi: 10.3934/dcds.2019081
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Gilbarg and N. Turdinger, Elliptic Partical Differential Equations of Second Order, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Turing, The chemical basis of morphogenesis, Phil. Trans.Roy, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

D. Gilbarg and N. Turdinger, Elliptic Partical Differential Equations of Second Order, Springer, Berlin, 1983. doi: 10.1007/978-3-642-61798-0.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[11]

A. Turing, The chemical basis of morphogenesis, Phil. Trans.Roy, 237 (1952), 37-72.  doi: 10.1098/rstb.1952.0012.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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