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Klein-Gordon equations on modulation spaces. (English) Zbl 1474.35484

Summary: We consider the Cauchy problem for a family of Klein-Gordon equations with initial data in modulation spaces \(M_{p, 1}^a\). We develop the well-posedness, blowup criterion, stability of regularity, scattering theory, and stability theory.

MSC:

35L71 Second-order semilinear hyperbolic equations
35L15 Initial value problems for second-order hyperbolic equations
35P25 Scattering theory for PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35B35 Stability in context of PDEs
35B44 Blow-up in context of PDEs
35B65 Smoothness and regularity of solutions to PDEs
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References:

[1] Ruzhansky, M.; Sugimoto, M.; Wang, B. X., Modulation spaces and nonlinear evolution equations, Progress in Mathematics, 301, 267-283 (2012) · Zbl 1256.42038
[2] Wang, B. X.; Hudzik, H., The global Cauchy problem for the NLS and NLKG with small rough data, Journal of Differential Equations, 232, 1, 36-73 (2007) · Zbl 1121.35132
[3] Guo, W.; Chen, J., Stability of schrödinger equations on modulation spaces, Frontiers of Mathematics in China, 9, 2, 275-301 (2014) · Zbl 1311.42058
[4] Bényi, Á.; Okoudjou, K. A., Local well-posedness of nonlinear dispersive equations on modulation spaces, Bulletin of the London Mathematical Society, 41, 3, 549-558 (2009) · Zbl 1173.35115 · doi:10.1112/blms/bdp027
[5] Tao, T., Nonlinear dispersive equations: local and global analysis · Zbl 1106.35001
[6] Feichtinger, H. G., Modulation spaces on locally compact Abelian group (1983), University of Vienna
[7] Wang, B. X.; Zhao, L. F.; Guo, B. L., Isometric decomposition operators, function spaces \(E_{p, q}^\lambda\) and applications to nonlinear evolution equations, Journal of Functional Analysis, 233, 1, 1-39 (2006) · Zbl 1099.46023 · doi:10.1016/j.jfa.2005.06.018
[8] Triebel, H., Theory of Function Spaces (1983), Basel, Switzerland: Birkhauser, Basel, Switzerland · Zbl 0546.46028
[9] Bényi, Á.; Gröchenig, K.; Okoudjou, K. A.; Rogers, L. G., Unimodular Fourier multipliers for modulation spaces, Journal of Functional Analysis, 246, 2, 366-384 (2007) · Zbl 1120.42010 · doi:10.1016/j.jfa.2006.12.019
[10] Wang, B.; Hao, C.; Huo, C., Harmonic Analysis Method For Nonlinear Evolution Equations I (2011), Hackensack, NJ, USA: World Scientific, Hackensack, NJ, USA · Zbl 1254.35002
[11] Chen, J.; Fan, D., Estimates for wave and Klein-Gordon equations on modulation sapces, Science China Mathematics, 55, 10, 2109-2123 (2012) · Zbl 1258.42022
[12] Strauss, W. A., Nonlinear scattering theory at low energy, Journal of Functional Analysis, 41, 1, 110-133 (1981) · Zbl 0466.47006 · doi:10.1016/0022-1236(81)90063-X
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