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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. K. Bhowmik, "Numerical Approximation of a Nonlinear Partial Integro-Differential Equation," PhD thesis, Heriot-Watt University, Edinburgh, UK, April, 2008. Google Scholar

[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.  Google Scholar

[6]

A. Chmaj and X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electr. J. Diff.eqns., 2002 (2002), 1-12.  Google Scholar

[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.  Google Scholar

[8]

K. Deng, On a nonlocal reaction-diffusion population model, DCDS series B., 9 (2008), 65-73.  Google Scholar

[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.  Google Scholar

[10]

P. C. Fife, Models of phase separation and their mathematics, Electronic Journal of Differential Equations, 48 (2000), 1-26. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory: Vol. II," Springer-Verlag, New York, 1988.  Google Scholar

[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.  Google Scholar

[15]

W. J. F. Govaerts, "Numerical Methods for Bifurcations of Dynamical Equilibria," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719543.  Google Scholar

[16]

M. Grinfeld, W. Hines, V. Hutson, K. Mischaikow and G. Vickers, Non-local dispersal, Differential and Integral Equations, 11 (2005), 1299-1320.  Google Scholar

[17]

M. Grinfeld and I. Stoleriu, Truncated gradient flows of the van der Waals free energy, Electron. J. Diff. Eqns., 2006 (2006), 1-9.  Google Scholar

[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.  Google Scholar

[19]

R. B. Hoyle, "Pattern Formation: An Introduction to Methods," Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511616051.  Google Scholar

[20]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. Journal of Applied Mathematics, 17 (2006), 211-232. doi: 10.1017/S0956792506006462.  Google Scholar

[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.  Google Scholar

[22]

Z. Mei, "Numerical Bifurcation Analysis for Reaction-Diffusion Equations," Springer, 2000.  Google Scholar

[23]

K. E. Morrison, Spectral approximation of multiplication operators, New York Journal of Mathematics, 1 (1995), 75-96.  Google Scholar

[24]

J. D. Rossi and A. F. Pazoto, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Analysis, 52 (2007), 143-155.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

S. K. Bhowmik, "Numerical Approximation of a Nonlinear Partial Integro-Differential Equation," PhD thesis, Heriot-Watt University, Edinburgh, UK, April, 2008. Google Scholar

[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.  Google Scholar

[6]

A. Chmaj and X. Ren, The nonlocal bistable equation: stationary solutions on a bounded interval, Electr. J. Diff.eqns., 2002 (2002), 1-12.  Google Scholar

[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.  Google Scholar

[8]

K. Deng, On a nonlocal reaction-diffusion population model, DCDS series B., 9 (2008), 65-73.  Google Scholar

[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.  Google Scholar

[10]

P. C. Fife, Models of phase separation and their mathematics, Electronic Journal of Differential Equations, 48 (2000), 1-26. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

M. Golubitsky, I. N. Stewart and D. G. Schaeffer, "Singularities and Groups in Bifurcation Theory: Vol. II," Springer-Verlag, New York, 1988.  Google Scholar

[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.  Google Scholar

[15]

W. J. F. Govaerts, "Numerical Methods for Bifurcations of Dynamical Equilibria," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719543.  Google Scholar

[16]

M. Grinfeld, W. Hines, V. Hutson, K. Mischaikow and G. Vickers, Non-local dispersal, Differential and Integral Equations, 11 (2005), 1299-1320.  Google Scholar

[17]

M. Grinfeld and I. Stoleriu, Truncated gradient flows of the van der Waals free energy, Electron. J. Diff. Eqns., 2006 (2006), 1-9.  Google Scholar

[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.  Google Scholar

[19]

R. B. Hoyle, "Pattern Formation: An Introduction to Methods," Cambridge University Press, Cambridge, 2006. doi: 10.1017/CBO9780511616051.  Google Scholar

[20]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability, Euro. Journal of Applied Mathematics, 17 (2006), 211-232. doi: 10.1017/S0956792506006462.  Google Scholar

[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.  Google Scholar

[22]

Z. Mei, "Numerical Bifurcation Analysis for Reaction-Diffusion Equations," Springer, 2000.  Google Scholar

[23]

K. E. Morrison, Spectral approximation of multiplication operators, New York Journal of Mathematics, 1 (1995), 75-96.  Google Scholar

[24]

J. D. Rossi and A. F. Pazoto, Asymptotic behaviour for a semilinear nonlocal equation, Asymptotic Analysis, 52 (2007), 143-155.  Google Scholar

[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.  Google Scholar

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