×

Turing patterns in general reaction-diffusion systems of Brusselator type. (English) Zbl 1198.35276

Summary: We study the reaction-diffusion system
\[ \begin{cases} u_t-d_1\Delta u=a-(b+1)u+f(u)v &\text{in }\Omega\times(0,T),\\ v_t-d_2\Delta v=bu-f(u)v &\text{in }\Omega\times(0,T),\\ u(x,0)=u_0(x) \quad v(x,0)= v_0(x) &\text{on }\Omega,\\ \frac{\partial u}{\partial\nu}(x,t)= \frac{\partial u}{\partial\nu}(x,t)=0 &\text{on }\partial\Omega\times(0,T), \end{cases} \]
Here, \(\Omega\) is a smooth and bounded domain in \(\mathbb R^N\) \((N\geq 1)\), \(a,b,d_1,d_2>0\) and \(f\in C^1[0,\infty)\) is a non-decreasing function. The case \(f(u)=u^2\)A corresponds to the standard Brusselator model for autocatalytic oscillating chemical reactions. Our analysis points out the crucial role played by the nonlinearity \(f\) in the existence of Turing patterns. More precisely, we show that, if \(f\) has a sublinear growth, then no Turing patterns occur, while if \(f\) has a superlinear growth, then existence of such patterns is strongly related to the inter-dependence between the parameters \(a, b\) and the diffusion coefficients \(d_1,d_2\).

MSC:

35Q92 PDEs in connection with biology, chemistry and other natural sciences
92E20 Classical flows, reactions, etc. in chemistry
92C15 Developmental biology, pattern formation
47H11 Degree theory for nonlinear operators
58C15 Implicit function theorems; global Newton methods on manifolds
35B35 Stability in context of PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] DOI: 10.1088/0951-7715/8/5/004 · Zbl 0839.35010 · doi:10.1088/0951-7715/8/5/004
[2] Brezis H., Analyse Fonctionnelle. Théorie et Applications (1983)
[3] DOI: 10.1007/978-1-4612-4126-3_2 · doi:10.1007/978-1-4612-4126-3_2
[4] DOI: 10.1016/0362-546X(94)00218-7 · Zbl 0829.35010 · doi:10.1016/0362-546X(94)00218-7
[5] DOI: 10.1137/0143082 · Zbl 0529.92029 · doi:10.1137/0143082
[6] DOI: 10.1063/1.1681288 · doi:10.1063/1.1681288
[7] DOI: 10.1088/0951-7715/21/10/007 · Zbl 1156.35022 · doi:10.1088/0951-7715/21/10/007
[8] DOI: 10.1016/0009-2509(83)80132-8 · doi:10.1016/0009-2509(83)80132-8
[9] Henry D., Lectures Notes in Mathematics, No. 840, in: Geometric Theory of Semilinear Parabolic Equations (1993)
[10] DOI: 10.3934/dcds.2008.20.939 · Zbl 1160.37036 · doi:10.3934/dcds.2008.20.939
[11] DOI: 10.1016/j.physd.2005.12.005 · Zbl 1108.35088 · doi:10.1016/j.physd.2005.12.005
[12] DOI: 10.1126/science.251.4994.650 · doi:10.1126/science.251.4994.650
[13] DOI: 10.1006/jdeq.1996.0157 · Zbl 0867.35032 · doi:10.1006/jdeq.1996.0157
[14] DOI: 10.1006/jdeq.1998.3559 · Zbl 0934.35040 · doi:10.1006/jdeq.1998.3559
[15] DOI: 10.1090/cln/006 · doi:10.1090/cln/006
[16] DOI: 10.1103/PhysRevE.64.056213 · doi:10.1103/PhysRevE.64.056213
[17] DOI: 10.1088/0951-7715/21/7/006 · Zbl 1148.35094 · doi:10.1088/0951-7715/21/7/006
[18] DOI: 10.1016/j.jmaa.2004.12.026 · Zbl 1108.35049 · doi:10.1016/j.jmaa.2004.12.026
[19] Prigogine I., J. Chem. Phys. 48 pp 1665–
[20] Rothe F., Lecture Notes in Mathematics, in: Global Solutions in Reaction-Diffusion Systems (1983)
[21] DOI: 10.1016/0022-5193(79)90042-0 · doi:10.1016/0022-5193(79)90042-0
[22] DOI: 10.1111/j.1432-1033.1968.tb00175.x · doi:10.1111/j.1432-1033.1968.tb00175.x
[23] DOI: 10.1007/978-1-4612-0873-0 · doi:10.1007/978-1-4612-0873-0
[24] DOI: 10.1098/rstb.1952.0012 · Zbl 1403.92034 · doi:10.1098/rstb.1952.0012
[25] DOI: 10.1023/A:1019158500612 · Zbl 1016.92049 · doi:10.1023/A:1019158500612
[26] DOI: 10.1017/S0308210500000627 · Zbl 0971.35040 · doi:10.1017/S0308210500000627
[27] DOI: 10.4310/DPDE.2007.v4.n2.a4 · Zbl 1158.37028 · doi:10.4310/DPDE.2007.v4.n2.a4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.