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2015, 2015(special): 901-905. doi: 10.3934/proc.2015.0901

Remarks on a dispersive equation in de Sitter spacetime

Makoto Nakamura 1,

1. 

Faculty of Science, Yamagata University, Kojirakawa-machi 1-4-12, Yamagata 990-8560

Received  September 2014 Revised  February 2015 Published  November 2015

Some nonlinear Schrödinger equations are derived from the nonrelativistic limit of nonlinear Klein-Gordon equations in de Sitter spacetime. Time local solutions for the Cauchy problem are considered in Sobolev spaces for power type nonlinear terms. The roles of spatial expansion and contraction on the problem are studied.
Keywords: Cauchy Problem, Nonlinear Schrödinger equation, de Sitter spacetime..
Mathematics Subject Classification: Primary: 35Q55; Secondary: 35L7.
Citation: Makoto Nakamura. Remarks on a dispersive equation in de Sitter spacetime. Conference Publications, 2015, 2015 (special) : 901-905. doi: 10.3934/proc.2015.0901
References:
[1]

V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations, 32 (2007), no. 10-12, 1643-1677.

[2]

D. Baskin, A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces, J. Funct. Anal., 259 (2010), no. 7, 1673-1719.

[3]

T. Cazenave, "Semilinear Schrödinger equations,'' Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.

[4]

G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), no. 3, 207-238.

[5]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), no. 1, 50-68.

[6]

A. D. Ionescu, B. Pausader and G. Staffilani, On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE, 5 (2012), no. 4, 705-746.

[7]

J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma, Phys. Fluids, 20 (1977), 1176-1179.

[8]

M. Nakamura, The Cauchy problem for semi-linear Klein-Gordon equations in de Sitter spacetime, J. Math. Anal. Appl., 410 (2014), no. 1, 445-454.

[9]

M. Nakamura and T. Ozawa, Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Rev. Math. Phys., 9 (1997), no. 3, 397-410.

[10]

T. Tao, "Nonlinear dispersive equations. Local and global analysis," CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373.

[11]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), no. 1, 115-125.

[12]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc. (N.S.), 11 (1984), no. 1, 186-188.

show all references

References:
[1]

V. Banica, The nonlinear Schrödinger equation on hyperbolic space, Comm. Partial Differential Equations, 32 (2007), no. 10-12, 1643-1677.

[2]

D. Baskin, A parametrix for the fundamental solution of the Klein-Gordon equation on asymptotically de Sitter spaces, J. Funct. Anal., 259 (2010), no. 7, 1673-1719.

[3]

T. Cazenave, "Semilinear Schrödinger equations,'' Courant Lecture Notes in Mathematics, 10, New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2003. xiv+323 pp.

[4]

G. B. Folland and A. Sitaram, The uncertainty principle: a mathematical survey, J. Fourier Anal. Appl., 3 (1997), no. 3, 207-238.

[5]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), no. 1, 50-68.

[6]

A. D. Ionescu, B. Pausader and G. Staffilani, On the global well-posedness of energy-critical Schrödinger equations in curved spaces, Anal. PDE, 5 (2012), no. 4, 705-746.

[7]

J. F. Lam, B. Lippmann and F. Tappert, Self-trapped laser beams in plasma, Phys. Fluids, 20 (1977), 1176-1179.

[8]

M. Nakamura, The Cauchy problem for semi-linear Klein-Gordon equations in de Sitter spacetime, J. Math. Anal. Appl., 410 (2014), no. 1, 445-454.

[9]

M. Nakamura and T. Ozawa, Low energy scattering for nonlinear Schrödinger equations in fractional order Sobolev spaces, Rev. Math. Phys., 9 (1997), no. 3, 397-410.

[10]

T. Tao, "Nonlinear dispersive equations. Local and global analysis," CBMS Regional Conference Series in Mathematics, 106. Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2006. xvi+373.

[11]

Y. Tsutsumi, $L^2$-solutions for nonlinear Schrödinger equations and nonlinear groups, Funkcial. Ekvac., 30 (1987), no. 1, 115-125.

[12]

Y. Tsutsumi and K. Yajima, The asymptotic behavior of nonlinear Schrödinger equations, Bull. Amer. Math. Soc. (N.S.), 11 (1984), no. 1, 186-188.

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