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Existence of solution for a generalized stochastic Cahn-Hilliard equation on convex domains
Finite to infinite steady state solutions, bifurcations of an integro-differential equation
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 |
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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