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Weak attractor of the KleinGordon field in discrete spacetime interacting with a nonlinear oscillator
1.  Texas A&M University, College Station, Texas 77843, United States 
References:
[1] 
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," vol. 25 of Studies in Mathematics and its Applications, NorthHolland Publishing Co., Amsterdam, 1992, translated and revised from the 1989 Russian original by Babin. 
[2] 
J. Chabassier and P. Joly, Energy preserving schemes for nonlinear hamiltonian systems of wave equations. Application to the vibrating piano string, (2010), to appear in Comp. Methods in Appl. Mech. and Engineering. doi: 10.1016/j.cma.2010.04.013. 
[3] 
A. Comech and A. I. Komech, Wellposedness and the energy and charge conservation for nonlinear wave equations in discrete spacetime, Russ. J. Math. Phys., 18 (2011), 410419. doi: 10.1134/S1061920811040030. 
[4] 
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002. 
[5] 
M. S. Èskina, The scattering problem for partialdifference equations, in Math. Phys. No. 3 (1967) (Russian), 248273, Naukova Dumka, Kiev, 1967. 
[6] 
C. Foias, M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591613. 
[7] 
C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135153. 
[8] 
D. Furihata, Finitedifference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math., 134 (2001), 3757. doi: 10.1016/S03770427(00)005276. 
[9] 
L. Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer Study Edition, SpringerVerlag, Berlin, 1990, second edn. 
[10] 
S. Jiménez and L. Vázquez, Analysis of four numerical schemes for a nonlinear KleinGordon equation, Appl. Math. Comput., 35 (1990), 6194. 
[11] 
L. V. Kapitanskiĭ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Algebra i Analiz, 2 (1990), 114140. 
[12] 
A. I. Komech and A. A. Komech, Global attractor for a nonlinear oscillator coupled to the KleinGordon field, Arch. Ration. Mech. Anal., 185 (2007), 105142. doi: 10.1007/s002050060039z. 
[13] 
A. I. Komech and A. A. Komech, Global attraction to solitary waves for KleinGordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2008), 855868. 
[14] 
A. I. Komech and A. A. Komech, Global attractor for the KleinGordon field coupled to several nonlinear oscillators, J. Math. Pures Appl., 93 (2010), 91111. doi: 10.1016/j.matpur.2009.08.011. 
[15] 
A. I. Komech and A. A. Komech, On the Titchmarsh convolution theorem for distributions on a circle, Funktsional. Anal. i Prilozhen., 46 (2012), to appear (see arXiv:1108.2463). 
[16] 
E. A. Kopylova, Dispersive estimates for discrete Schrödinger and KleinGordon equations, Algebra i Analiz, 21 (2009), 87113. doi: 10.1090/S106100222010011154. 
[17] 
B. Y. Levin, "Lectures on Entire Functions," vol. 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996, in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko. 
[18] 
J.L. Lions, Supports de produits de composition. I, C. R. Acad. Sci. Paris, 232 (1951), 15301532. 
[19] 
S. Li and L. VuQuoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear KleinGordon equation, SIAM J. Numer. Anal., 32 (1995), 18391875. doi: 10.1137/0732083. 
[20] 
C. S. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math., 25 (1972), 131. 
[21] 
I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129135. 
[22] 
I. E. Segal, Nonlinear semigroups, Ann. of Math. (2), 78 (1963), 339364. 
[23] 
A. Soffer, Soliton dynamics and scattering, in "International Congress of Mathematicians. Vol. III," 459471, Eur. Math. Soc., Zürich, 2006. 
[24] 
W. A. Strauss, Decay and asymptotics for $\square u = f(u)$, J. Functional Analysis, 2 (1968), 409457. 
[25] 
W. Strauss and L. Vazquez, Numerical solution of a nonlinear KleinGordon equation, J. Comput. Phys., 28 (1978), 271278. 
[26] 
W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators, Appl. Anal., 80 (2001), 525556. doi: 10.1080/00036810108841007. 
[27] 
T. Tao, A (concentration)compact attractor for highdimensional nonlinear Schrödinger equations, Dyn. Partial Differ. Equ., 4 (2007), 153. 
[28] 
R. Temam, "InfiniteDimensional Dynamical Systems in Mechanics and Physics," vol. 68 of Applied Mathematical Sciences, SpringerVerlag, New York, 1997, second edn. 
[29] 
E. Titchmarsh, The zeros of certain integral functions, Proc. of the London Math. Soc., 25 (1926), 283302. 
[30] 
J. Virieux, PSV wave propagation in heterogeneous media: Velocitystress finitedifference method, Geophysics, 51 (1986), 889901. 
[31] 
K. Yosida, "Functional Analysis," vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], SpringerVerlag, Berlin, 1980, sixth edn. 
show all references
References:
[1] 
A. V. Babin and M. I. Vishik, "Attractors of Evolution Equations," vol. 25 of Studies in Mathematics and its Applications, NorthHolland Publishing Co., Amsterdam, 1992, translated and revised from the 1989 Russian original by Babin. 
[2] 
J. Chabassier and P. Joly, Energy preserving schemes for nonlinear hamiltonian systems of wave equations. Application to the vibrating piano string, (2010), to appear in Comp. Methods in Appl. Mech. and Engineering. doi: 10.1016/j.cma.2010.04.013. 
[3] 
A. Comech and A. I. Komech, Wellposedness and the energy and charge conservation for nonlinear wave equations in discrete spacetime, Russ. J. Math. Phys., 18 (2011), 410419. doi: 10.1134/S1061920811040030. 
[4] 
V. V. Chepyzhov and M. I. Vishik, "Attractors for Equations of Mathematical Physics," vol. 49 of American Mathematical Society Colloquium Publications, American Mathematical Society, Providence, RI, 2002. 
[5] 
M. S. Èskina, The scattering problem for partialdifference equations, in Math. Phys. No. 3 (1967) (Russian), 248273, Naukova Dumka, Kiev, 1967. 
[6] 
C. Foias, M. S. Jolly, I. G. Kevrekidis, and E. S. Titi, Dissipativity of numerical schemes, Nonlinearity, 4 (1991), 591613. 
[7] 
C. Foias and E. S. Titi, Determining nodes, finite difference schemes and inertial manifolds, Nonlinearity, 4 (1991), 135153. 
[8] 
D. Furihata, Finitedifference schemes for nonlinear wave equation that inherit energy conservation property, J. Comput. Appl. Math., 134 (2001), 3757. doi: 10.1016/S03770427(00)005276. 
[9] 
L. Hörmander, "The Analysis of Linear Partial Differential Operators. I," Springer Study Edition, SpringerVerlag, Berlin, 1990, second edn. 
[10] 
S. Jiménez and L. Vázquez, Analysis of four numerical schemes for a nonlinear KleinGordon equation, Appl. Math. Comput., 35 (1990), 6194. 
[11] 
L. V. Kapitanskiĭ and I. N. Kostin, Attractors of nonlinear evolution equations and their approximations, Algebra i Analiz, 2 (1990), 114140. 
[12] 
A. I. Komech and A. A. Komech, Global attractor for a nonlinear oscillator coupled to the KleinGordon field, Arch. Ration. Mech. Anal., 185 (2007), 105142. doi: 10.1007/s002050060039z. 
[13] 
A. I. Komech and A. A. Komech, Global attraction to solitary waves for KleinGordon equation with mean field interaction, Ann. Inst. H. Poincaré Anal. Non Linéaire, 26 (2008), 855868. 
[14] 
A. I. Komech and A. A. Komech, Global attractor for the KleinGordon field coupled to several nonlinear oscillators, J. Math. Pures Appl., 93 (2010), 91111. doi: 10.1016/j.matpur.2009.08.011. 
[15] 
A. I. Komech and A. A. Komech, On the Titchmarsh convolution theorem for distributions on a circle, Funktsional. Anal. i Prilozhen., 46 (2012), to appear (see arXiv:1108.2463). 
[16] 
E. A. Kopylova, Dispersive estimates for discrete Schrödinger and KleinGordon equations, Algebra i Analiz, 21 (2009), 87113. doi: 10.1090/S106100222010011154. 
[17] 
B. Y. Levin, "Lectures on Entire Functions," vol. 150 of Translations of Mathematical Monographs, American Mathematical Society, Providence, RI, 1996, in collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko, Translated from the Russian manuscript by Tkachenko. 
[18] 
J.L. Lions, Supports de produits de composition. I, C. R. Acad. Sci. Paris, 232 (1951), 15301532. 
[19] 
S. Li and L. VuQuoc, Finite difference calculus invariant structure of a class of algorithms for the nonlinear KleinGordon equation, SIAM J. Numer. Anal., 32 (1995), 18391875. doi: 10.1137/0732083. 
[20] 
C. S. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation, Comm. Pure Appl. Math., 25 (1972), 131. 
[21] 
I. E. Segal, The global Cauchy problem for a relativistic scalar field with power interaction, Bull. Soc. Math. France, 91 (1963), 129135. 
[22] 
I. E. Segal, Nonlinear semigroups, Ann. of Math. (2), 78 (1963), 339364. 
[23] 
A. Soffer, Soliton dynamics and scattering, in "International Congress of Mathematicians. Vol. III," 459471, Eur. Math. Soc., Zürich, 2006. 
[24] 
W. A. Strauss, Decay and asymptotics for $\square u = f(u)$, J. Functional Analysis, 2 (1968), 409457. 
[25] 
W. Strauss and L. Vazquez, Numerical solution of a nonlinear KleinGordon equation, J. Comput. Phys., 28 (1978), 271278. 
[26] 
W. Shaban and B. Vainberg, Radiation conditions for the difference Schrödinger operators, Appl. Anal., 80 (2001), 525556. doi: 10.1080/00036810108841007. 
[27] 
T. Tao, A (concentration)compact attractor for highdimensional nonlinear Schrödinger equations, Dyn. Partial Differ. Equ., 4 (2007), 153. 
[28] 
R. Temam, "InfiniteDimensional Dynamical Systems in Mechanics and Physics," vol. 68 of Applied Mathematical Sciences, SpringerVerlag, New York, 1997, second edn. 
[29] 
E. Titchmarsh, The zeros of certain integral functions, Proc. of the London Math. Soc., 25 (1926), 283302. 
[30] 
J. Virieux, PSV wave propagation in heterogeneous media: Velocitystress finitedifference method, Geophysics, 51 (1986), 889901. 
[31] 
K. Yosida, "Functional Analysis," vol. 123 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], SpringerVerlag, Berlin, 1980, sixth edn. 
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