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

The Schnakenberg model with precursors

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].
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. AoM. 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. BensonP. 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. GuiJ. 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. IronJ. 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. AoM. 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. BensonP. 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. GuiJ. 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. IronJ. 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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