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March  2016, 21(2): 641-654. doi: 10.3934/dcdsb.2016.21.641

Destabilization threshold curves for diffusion systems with equal diffusivity under non-diagonal flux boundary conditions

Kunimochi Sakamoto 1,

1. 

Graduate School of Science, Department of Mathematical and Life Sciences, Hiroshima University, 1-3-1 Kagamiyama, Higashi-Hiroshima, 739-8526, Japan

Received  December 2014 Revised  September 2015 Published  November 2015

This article deals with destabilizations of Turing type for diffusive systems with equal diffusivity under non-diagonal flux boundary conditions. Stability-instability threshold curves in the complex plane are described as the graph of a piecewise analytic function for simple $m$-dimensional domains $(m\geq 1)$. Also analyzed are effects caused by imposing homogeneous boundary conditions of Dirichlet or Neumann type on appropriate portions of the domain boundary.
Keywords: Stability, non-diagonal boundary conditions., reaction-diffusion equations, threshold.
Mathematics Subject Classification: Primary: 35K40, 35J25; Secondary: 35J1.
Citation: Kunimochi Sakamoto. Destabilization threshold curves for diffusion systems with equal diffusivity under non-diagonal flux boundary conditions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (2) : 641-654. doi: 10.3934/dcdsb.2016.21.641
References:
[1]

H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269. doi: 10.1016/0022-0396(88)90156-8.  Google Scholar

[2]

A. Anma and K. Sakamoto, Turing type mechanisms for linear diffusion systems under non-diagonal Robin boundary conditions, SIAM Journal on Mathematical Analysis, 45 (2013), 3611-3628. doi: 10.1137/130908270.  Google Scholar

[3]

J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33-59. doi: 10.1006/jdeq.2000.3876.  Google Scholar

[4]

G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numerical Func. Anal. Opt., 25 (2004), 321-348. doi: 10.1081/NFA-120039655.  Google Scholar

[5]

H. Levine and W.-J. Rappel, Membrane-bound Turing patterns, Physical Review E, 72 (2005), 061912, 5pp. doi: 10.1103/PhysRevE.72.061912.  Google Scholar

[6]

J. D. Murray, Mathematical Biology, Biomathematics Texts, Springer-Verlag Berlin Heidelberg, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[7]

Alan M. Turing, The chemical basis for morphogenesis, Phil. Trans. R. Soc. London, B 273 (1952), 37-72. Google Scholar

[8]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.  Google Scholar

show all references

References:
[1]

H. Amann, Parabolic evolution equations and nonlinear boundary conditions, J. Differential Equations, 72 (1988), 201-269. doi: 10.1016/0022-0396(88)90156-8.  Google Scholar

[2]

A. Anma and K. Sakamoto, Turing type mechanisms for linear diffusion systems under non-diagonal Robin boundary conditions, SIAM Journal on Mathematical Analysis, 45 (2013), 3611-3628. doi: 10.1137/130908270.  Google Scholar

[3]

J. M. Arrieta, A. N. Carvalho and A. Rodríguez-Bernal, Upper semicontinuity for attractors of parabolic problems with localized large diffusion and nonlinear boundary conditions, J. Differential Equations, 168 (2000), 33-59. doi: 10.1006/jdeq.2000.3876.  Google Scholar

[4]

G. Auchmuty, Steklov eigenproblems and the representation of solutions of elliptic boundary value problems, Numerical Func. Anal. Opt., 25 (2004), 321-348. doi: 10.1081/NFA-120039655.  Google Scholar

[5]

H. Levine and W.-J. Rappel, Membrane-bound Turing patterns, Physical Review E, 72 (2005), 061912, 5pp. doi: 10.1103/PhysRevE.72.061912.  Google Scholar

[6]

J. D. Murray, Mathematical Biology, Biomathematics Texts, Springer-Verlag Berlin Heidelberg, 1989. doi: 10.1007/978-3-662-08539-4.  Google Scholar

[7]

Alan M. Turing, The chemical basis for morphogenesis, Phil. Trans. R. Soc. London, B 273 (1952), 37-72. Google Scholar

[8]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge Mathematical Library. Cambridge University Press, Cambridge, 1995.  Google Scholar

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