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Multiplicity solutions for fully nonlinear equation involving nonlinearity with zeros
Global well-posedness in uniformly local spaces for the Cahn-Hilliard equation in $\mathbb{R}^3$
1. | Department of Mathematics, University of Surrey, Guildford, GU2 7XH |
2. | University of Surrey, Guildford, GU2 7XH, China |
References:
[1] |
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations, 83 (1990), 85-108.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
A. Babin, Global attractors in PDE. Handbook of dynamical systems, Vol. 1B, 983-1085, Elsevier B. V., Amsterdam, 2006.
doi: 10.1016/S1874-575X(06)80039-1. |
[3] |
A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Royal. Soc. Edimburgh, 116A (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[5] |
A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain, Quarterly of Applied Mathematics, 64 (2006), 94-104. |
[6] |
J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts, Comm. Pure Appl. Math., 52 (1999), 839-871.
doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. |
[7] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
[8] |
L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation, rch. Rational Mech. Anal., 133 (1995), 129-144.
doi: 10.1007/BF00376814. |
[9] |
L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
[10] |
A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymptotic Anal., 4 (1991), 161-185.
doi: 10.3233/ASY-1991-4202. |
[11] |
T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$, Journal Diff. Eqns., 252 (2012), 2771-2791.
doi: 10.1016/j.jde.2011.08.052. |
[12] |
A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation, Comm. Pure Appl. Anal., 6 (2007), 1075-1086.
doi: 10.3934/cpaa.2007.6.1075. |
[13] |
A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, ().
|
[14] |
C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems, Internat. Ser. Numer. Math., Birkhauser, Basel, 88 (1989), 35-73. |
[15] |
M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[16] |
M. Efendiev, H. Gajewski and S. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity, Adv. Differential Equations, 7 (2002), 1073-1100. |
[17] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.
doi: 10.1002/mana.200310186. |
[18] |
J. Evans, V. Galaktionov and J. Williams, Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity, SIAM Journal on Mathematical Analysis, 38 (2006), 64-102. |
[19] |
M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170.
doi: 10.1080/03605300802608247. |
[20] |
M. Grasselli, Maurizio, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404.
doi: 10.1007/s00028-009-0017-7. |
[21] |
M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60.
doi: 10.1016/j.jde.2007.05.003. |
[22] |
V. Kalantarov, Global behavior of the solutions of some fourth-order nonlinear equations. (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 163 (1987), Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsii 19, 66-75.
doi: 10.1007/BF02208712. |
[23] |
T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts, Comm. Pure Appl. Math., 58 (2005), 1077-1115. |
[24] |
A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. |
[25] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, 103-200, Handb. Differ. Equ., Elsevier North-Holland, Amsterdam, 2008.
doi: 10.1016/S1874-5717(08)00003-0. |
[26] |
A. Miranville and A. S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[27] |
A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.
doi: 10.1002/mma.464. |
[28] |
A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, Journal of Applied Analysis and Computation, 1 (2011), 523-536. |
[29] |
A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985. |
[30] |
A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys, Appl. Anal., 47 (1992), 241-257.
doi: 10.1080/00036819208840143. |
[31] |
Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases, Phys. Rev. Letters, 58 (1987), 836-839.
doi: 10.1103/PhysRevLett.58.836. |
[32] |
E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Phys. D, 192 (2004), 279-307.
doi: 10.1016/j.physd.2004.01.024. |
[33] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[34] |
J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459-492.
doi: 10.1016/S0294-1449(98)80031-0. |
[35] |
S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II, Int. Math. Ser. Springer, New York, (N. Y.), 7 (2008), 255-327.
doi: 10.1007/978-0-387-75219-8_6. |
[36] |
S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588.
doi: 10.1017/S0017089507003849. |
[37] |
S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains, J. Dynam. Differential Equations, 19 (2007), 1-74.
doi: 10.1007/s10884-006-9007-4. |
[38] |
S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637.
doi: 10.1002/cpa.10068. |
show all references
References:
[1] |
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations, 83 (1990), 85-108.
doi: 10.1016/0022-0396(90)90070-6. |
[2] |
A. Babin, Global attractors in PDE. Handbook of dynamical systems, Vol. 1B, 983-1085, Elsevier B. V., Amsterdam, 2006.
doi: 10.1016/S1874-575X(06)80039-1. |
[3] |
A. V. Babin, M. I. Vishik, "Attractors of Evolution Equations," Studies in Mathematics and its Applications, 25. North-Holland Publishing Co., Amsterdam, 1992. |
[4] |
A. V. Babin and M. I. Vishik, Attractors of partial differential equations in an unbounded domain, Proc. Royal. Soc. Edimburgh, 116A (1990), 221-243.
doi: 10.1017/S0308210500031498. |
[5] |
A. Bonfoh, Finite-dimensional attractor for the viscious Cahn-Hilliard equation in an unbounded domain, Quarterly of Applied Mathematics, 64 (2006), 94-104. |
[6] |
J. Bricmont, A. Kupiainen and J. Taskinen, Stability of Cahn-Hilliard fronts, Comm. Pure Appl. Math., 52 (1999), 839-871.
doi: 10.1002/(SICI)1097-0312(199907)52:7<839::AID-CPA4>3.0.CO;2-I. |
[7] |
J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.
doi: 10.1063/1.1744102. |
[8] |
L. Caffarelli and N. Muler, An $L^\infty$ bound for solutions of the Cahn-Hilliard equation, rch. Rational Mech. Anal., 133 (1995), 129-144.
doi: 10.1007/BF00376814. |
[9] |
L. Cherfils, A. Miranville and S. Zelik, The Cahn-Hilliard equation with logarithmic potentials, Milan J. Math., 79 (2011), 561-596.
doi: 10.1007/s00032-011-0165-4. |
[10] |
A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymptotic Anal., 4 (1991), 161-185.
doi: 10.3233/ASY-1991-4202. |
[11] |
T. Dlotko, M. Kania and C. Sun, Analysis of the viscous Cahn-Hilliard equation in $\R^N$, Journal Diff. Eqns., 252 (2012), 2771-2791.
doi: 10.1016/j.jde.2011.08.052. |
[12] |
A. Eden and V. K. Kalantarov, 3D convective Cahn - Hilliard equation, Comm. Pure Appl. Anal., 6 (2007), 1075-1086.
doi: 10.3934/cpaa.2007.6.1075. |
[13] |
A. Eden, V. Kalantarov and S. Zelik, Infinite-energy solutions for the Cahn-Hilliard equation in cylindrical domains,, submitted. \arXiv{1005.3424}, ().
|
[14] |
C. Elliott, The Cahn-Hilliard model for the kinetics of phase separation. Mathematical models for phase change problems, Internat. Ser. Numer. Math., Birkhauser, Basel, 88 (1989), 35-73. |
[15] |
M. Efendiev and S. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math., 54 (2001), 625-688.
doi: 10.1002/cpa.1011. |
[16] |
M. Efendiev, H. Gajewski and S. Zelik, The finite dimensional attractor for a 4th order system of Cahn-Hilliard type with a supercritical nonlinearity, Adv. Differential Equations, 7 (2002), 1073-1100. |
[17] |
M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a singularly perturbed Cahn-Hilliard system, Math. Nachr., 272 (2004), 11-31.
doi: 10.1002/mana.200310186. |
[18] |
J. Evans, V. Galaktionov and J. Williams, Blow-up and global asymptotics of the unstable Cahn-Hilliard equation with a homogeneous nonlinearity, SIAM Journal on Mathematical Analysis, 38 (2006), 64-102. |
[19] |
M. Grasselli, G. Schimperna and S. Zelik, On the 2D Cahn-Hilliard equation with inertial term, Comm. Partial Differential Equations, 34 (2009), 137-170.
doi: 10.1080/03605300802608247. |
[20] |
M. Grasselli, Maurizio, G. Schimperna, A. Segatti and S. Zelik, On the 3D Cahn-Hilliard equation with inertial term, J. Evol. Equ., 9 (2009), 371-404.
doi: 10.1007/s00028-009-0017-7. |
[21] |
M. Grasselli, H. Petzeltova and G. Schimperna, Asymptotic behavior of a nonisothermal viscous Cahn-Hilliard equation with inertial term, J. Differential Equations, 239 (2007), 38-60.
doi: 10.1016/j.jde.2007.05.003. |
[22] |
V. Kalantarov, Global behavior of the solutions of some fourth-order nonlinear equations. (Russian), Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 163 (1987), Kraev. Zadachi Mat. Fiz. i Smezhn. Vopr. Teor. Funktsii 19, 66-75.
doi: 10.1007/BF02208712. |
[23] |
T. Korvola, A. Kupiainen and J. Taskinen, Anomalous scaling for 3D Cahn-Hilliard Fronts, Comm. Pure Appl. Math., 58 (2005), 1077-1115. |
[24] |
A. Miranville and S. Zelik, Doubly nonlinear Cahn-Hilliard-Gurtin equations, Hokkaido Math. J., 38 (2009), 315-360. |
[25] |
A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in "Handbook of Differential Equations: Evolutionary Equations," Vol. IV, 103-200, Handb. Differ. Equ., Elsevier North-Holland, Amsterdam, 2008.
doi: 10.1016/S1874-5717(08)00003-0. |
[26] |
A. Miranville and A. S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.
doi: 10.1002/mma.590. |
[27] |
A. Miranville and S. Zelik, Robust exponential attractors for Cahn-Hilliard type equations with singular potentials, Math. Methods Appl. Sci., 27 (2004), 545-582.
doi: 10.1002/mma.464. |
[28] |
A. Miranville, Asymptotic behavior of the Cahn-Hilliard-Oono equation, Journal of Applied Analysis and Computation, 1 (2011), 523-536. |
[29] |
A. Novick-Cohen, The Cahn-Hilliard equation: mathematical and modeling perspectives, Adv. Math. Sci. Appl., 8 (1998), 965-985. |
[30] |
A. Novick-Cohen, Blow up and growth in the directional solidification of dilute binary alloys, Appl. Anal., 47 (1992), 241-257.
doi: 10.1080/00036819208840143. |
[31] |
Y. Oono and S. Puri, Computionally efficient modeling of ordering of quenched phases, Phys. Rev. Letters, 58 (1987), 836-839.
doi: 10.1103/PhysRevLett.58.836. |
[32] |
E. Rocca and G. Schimperna, Universal attractor for some singular phase transition systems, Phys. D, 192 (2004), 279-307.
doi: 10.1016/j.physd.2004.01.024. |
[33] |
R. Temam, "Infinite-dimensional Dynamical Systems in Mechanics and Physics," Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. |
[34] |
J. Wei and M. Winter, Stationary solutions for the Cahn-Hilliard equation, Ann. Inst. H. Poincar Anal. Non Lineaire, 15 (1998), 459-492.
doi: 10.1016/S0294-1449(98)80031-0. |
[35] |
S. Zelik, Weak spatially nondecaying solutions of 3D Navier-Stokes equations in cylindrical domains. Instability in models connected with fluid flows. II, Int. Math. Ser. Springer, New York, (N. Y.), 7 (2008), 255-327.
doi: 10.1007/978-0-387-75219-8_6. |
[36] |
S. Zelik, Spatially nondecaying solutions of the 2D Navier-Stokes equation in a strip, Glasg. Math. J., 49 (2007), 525-588.
doi: 10.1017/S0017089507003849. |
[37] |
S. Zelik, Spatial and dynamical chaos generated by reaction-diffusion systems in unbounded domains, J. Dynam. Differential Equations, 19 (2007), 1-74.
doi: 10.1007/s10884-006-9007-4. |
[38] |
S. Zelik, Attractors of reaction-diffusion systems in unbounded domains and their spatial complexity, Comm. Pure Appl. Math., 56 (2003), 584-637.
doi: 10.1002/cpa.10068. |
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