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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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[4]

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

[5]

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

[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. Google Scholar

[7]

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

[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. Google Scholar

[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. Google Scholar

[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. Google Scholar

[11]

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

[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. Google Scholar

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