Finite to infinite steady state solutions, bifurcations of an integro-differential equation

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July 2011, 16(1): 57-71. doi: 10.3934/dcdsb.2011.16.57

Finite to infinite steady state solutions, bifurcations of an integro-differential equation

Samir K. Bhowmik ^1, , Dugald B. Duncan ^2, , Michael Grinfeld ^3, and Gabriel J. Lord ^2,

1.	Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh
2.	Department of Mathematics and Maxwell Institute, Heriot-Watt University, Edinburgh, United Kingdom, United Kingdom
3.	Department of Mathematics, University of Strathclyde, Glasgow, United Kingdom

Received April 2010 Revised November 2010 Published April 2011

Abstract
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We consider a bistable integral equation which governs the stationary solutions of a convolution model of solid--solid phase transitions on a circle. We study the bifurcations of the set of the stationary solutions as the diffusion coefficient is increased to examine the transition from an uncountably infinite number of steady states to three for the continuum limit of the semi--discretised system. We show how the symmetry of the problem is responsible for the generation and stabilisation of equilibria and comment on the puzzling connection between continuity and stability that exists in this problem.

Keywords: continuum limit, bifurcations, regularity., Integro-differential equation, semi-discrete.

Mathematics Subject Classification: Primary: 37G40; Secondary: 45K0.

Citation: Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete and Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57

References:

[1]	P. Bates and F. Chen, Periodic travelling waves for a nonlocal integro-differential model, Electronic Journal of Differential Equations, 1999 (1999), 1-19.
[2]	P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189.
[3]	P. W. Bates, P. C. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138 (1997), 105-136. doi: 10.1007/s002050050037.
[4]	S. K. Bhowmik, "Numerical Approximation of a Nonlinear Partial Integro-Differential Equation," PhD thesis, Heriot-Watt University, Edinburgh, UK, April, 2008.
[5]	F. Chen, Uniform stability of multidimensional travelling waves for the nonlocal Allen-Cahn equation, Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference, 10 (2003), 109-113.
[6]	A. Chmaj and X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electr. J. Diff.eqns., 2002 (2002), 1-12.
[7]	J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Analysis, 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.
[8]	K. Deng, On a nonlocal reaction-diffusion population model, DCDS series B., 9 (2008), 65-73.
[9]	D. B. Duncan, M. Grinfeld and I. Stoleriu, Coarsening in an integro-differential model of phase transitions, Euro. Journal of Applied Mathematics, 11 (2000), 511-523. doi: 10.1017/S0956792500004319.
[10]	P. C. Fife, Models of phase separation and their mathematics, Electronic Journal of Differential Equations, 48 (2000), 1-26.
[11]	P. C. Fife, Well-posedness issues for models of phase transitions with weak interaction, Nonlinearity, 14 (2001), 221-238. doi: 10.1088/0951-7715/14/2/303.
[12]	J. Garcia Melian and J. D. Rossi, Logistic equation with refuge and nonlocal diffusion, Communications on Pure and Applied Analysis, 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037.
[13]	M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory: Vol. II," Springer-Verlag, New York, 1988.
[14]	W. J. F. Govaerts, Numerical bifurcation analysis for {ODEs}, Journal of Computational and Applied Mathematics, 125 (2000), 57-68. doi: 10.1016/S0377-0427(00)00458-1.
[15]	W. J. F. Govaerts, "Numerical Methods for Bifurcations of Dynamical Equilibria," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719543.
[16]	M. Grinfeld, W. Hines, V. Hutson, K. Mischaikow and G. Vickers, Non-local dispersal, Differential and Integral Equations, 11 (2005), 1299-1320.
[17]	M. Grinfeld and I. Stoleriu, Truncated gradient flows of the van der Waals free energy, Electron. J. Diff. Eqns., 2006 (2006), 1-9.
[18]	T. Hartley and T. Wanner, A semi-implicit spectral method for stochastic nonlocal phase-field models, DCDS, 25 (2009), 399-429. doi: 10.3934/dcds.2009.25.399.
[19]	R. B. Hoyle, "Pattern Formation: An Introduction to Methods," Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511616051.
[20]	V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. Journal of Applied Mathematics, 17 (2006), 211-232. doi: 10.1017/S0956792506006462.
[21]	J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Mathematical Biosciences, 184 (2003), 201-222. doi: 10.1016/S0025-5564(03)00041-5.
[22]	Z. Mei, "Numerical Bifurcation Analysis for Reaction-Diffusion Equations," Springer, 2000.
[23]	K. E. Morrison, Spectral approximation of multiplication operators, New York Journal of Mathematics, 1 (1995), 75-96.
[24]	J. D. Rossi and A. F. Pazoto, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Analysis, 52 (2007), 143-155.
[25]	H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5.

show all references

References:

[1]	P. Bates and F. Chen, Periodic travelling waves for a nonlocal integro-differential model, Electronic Journal of Differential Equations, 1999 (1999), 1-19.
[2]	P. W. Bates and A. Chmaj, A discrete convolution model for phase transitions, Arch. Ration. Mech. Anal., 150 (1999), 281-305. doi: 10.1007/s002050050189.
[3]	P. W. Bates, P. C. Fife, X. Ren and X. Wang, Travelling waves in a convolution model for phase transitions, Archive for Rational Mechanics and Analysis, 138 (1997), 105-136. doi: 10.1007/s002050050037.
[4]	S. K. Bhowmik, "Numerical Approximation of a Nonlinear Partial Integro-Differential Equation," PhD thesis, Heriot-Watt University, Edinburgh, UK, April, 2008.
[5]	F. Chen, Uniform stability of multidimensional travelling waves for the nonlocal Allen-Cahn equation, Fifth Mississippi State Conference on Differential Equations and Computational Simulations, Electronic Journal of Differential Equations, Conference, 10 (2003), 109-113.
[6]	A. Chmaj and X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electr. J. Diff.eqns., 2002 (2002), 1-12.
[7]	J. Coville and L. Dupaigne, Propagation speed of travelling fronts in non local reaction-diffusion equations, Nonlinear Analysis, 60 (2005), 797-819. doi: 10.1016/j.na.2003.10.030.
[8]	K. Deng, On a nonlocal reaction-diffusion population model, DCDS series B., 9 (2008), 65-73.
[9]	D. B. Duncan, M. Grinfeld and I. Stoleriu, Coarsening in an integro-differential model of phase transitions, Euro. Journal of Applied Mathematics, 11 (2000), 511-523. doi: 10.1017/S0956792500004319.
[10]	P. C. Fife, Models of phase separation and their mathematics, Electronic Journal of Differential Equations, 48 (2000), 1-26.
[11]	P. C. Fife, Well-posedness issues for models of phase transitions with weak interaction, Nonlinearity, 14 (2001), 221-238. doi: 10.1088/0951-7715/14/2/303.
[12]	J. Garcia Melian and J. D. Rossi, Logistic equation with refuge and nonlocal diffusion, Communications on Pure and Applied Analysis, 8 (2009), 2037-2053. doi: 10.3934/cpaa.2009.8.2037.
[13]	M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory: Vol. II," Springer-Verlag, New York, 1988.
[14]	W. J. F. Govaerts, Numerical bifurcation analysis for {ODEs}, Journal of Computational and Applied Mathematics, 125 (2000), 57-68. doi: 10.1016/S0377-0427(00)00458-1.
[15]	W. J. F. Govaerts, "Numerical Methods for Bifurcations of Dynamical Equilibria," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719543.
[16]	M. Grinfeld, W. Hines, V. Hutson, K. Mischaikow and G. Vickers, Non-local dispersal, Differential and Integral Equations, 11 (2005), 1299-1320.
[17]	M. Grinfeld and I. Stoleriu, Truncated gradient flows of the van der Waals free energy, Electron. J. Diff. Eqns., 2006 (2006), 1-9.
[18]	T. Hartley and T. Wanner, A semi-implicit spectral method for stochastic nonlocal phase-field models, DCDS, 25 (2009), 399-429. doi: 10.3934/dcds.2009.25.399.
[19]	R. B. Hoyle, "Pattern Formation: An Introduction to Methods," Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511616051.
[20]	V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. Journal of Applied Mathematics, 17 (2006), 211-232. doi: 10.1017/S0956792506006462.
[21]	J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new, Mathematical Biosciences, 184 (2003), 201-222. doi: 10.1016/S0025-5564(03)00041-5.
[22]	Z. Mei, "Numerical Bifurcation Analysis for Reaction-Diffusion Equations," Springer, 2000.
[23]	K. E. Morrison, Spectral approximation of multiplication operators, New York Journal of Mathematics, 1 (1995), 75-96.
[24]	J. D. Rossi and A. F. Pazoto, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Analysis, 52 (2007), 143-155.
[25]	H. R. Wilson and J. D. Cowan, Excitatory and inhibitory interactions in localized populations of model neurons, Biophys. J, 12 (1972), 1-24. doi: 10.1016/S0006-3495(72)86068-5.

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