## Long time decay for 2D Klein-Gordon equation.(English)Zbl 1192.35151

Summary: We obtain a dispersive long-time decay in weighted energy norms for solutions of the 2D Klein-Gordon equations. The decay extends the results obtained by Jensen, Kato and Murata for the equations of Schrödinger type by the spectral approach. For the proof, we modify the approach to make it applicable to relativistic equations.

### MSC:

 35Q53 KdV equations (Korteweg-de Vries equations) 35Q40 PDEs in connection with quantum mechanics 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 35B40 Asymptotic behavior of solutions to PDEs
Full Text:

### References:

 [1] Agmon, S., Spectral properties of Schrödinger operator and scattering theory, Ann. sc. norm. super. Pisa, ser. IV, 2, 151-218, (1975) · Zbl 0315.47007 [2] Brenner, P., On scattering and everywhere defined scattering operators for nonlinear klein – gordon equations, J. differential equations, 56, 310-344, (1985) · Zbl 0513.35066 [3] Buslaev, V.S.; Perelman, G., On the stability of solitary waves for nonlinear Schrödinger equations, Trans. amer. math. soc., 164, 75-98, (1995) · Zbl 0841.35108 [4] Buslaev, V.S.; Sulem, C., On asymptotic stability of solitary waves for nonlinear Schrödinger equations, Ann. inst. H. Poincaré anal. non linéaire, 20, 3, 419-475, (2003) · Zbl 1028.35139 [5] Cuccagna, S., Stabilization of solutions to nonlinear Schrödinger equations, Comm. pure appl. math., 54, 9, 1110-1145, (2001) · Zbl 1031.35129 [6] Delort, J.-M., Global existence and asymptotics for the quasilinear klein – gordon equation with small data in one space dimension, Ann. sci. école norm. sup. (4), 34, 1, 1-61, (2001), (in French) · Zbl 0990.35119 [7] Imaikin, V.; Komech, A.; Vainberg, B., On scattering of solitons for the klein – gorgon equation coupled to a particle, Comm. math. phys., 268, 2, 321-367, (2006) · Zbl 1127.35054 [8] Jensen, A., Spectral properties of Schrödinger operators and time-decay of the wave function. results in $$L^2(\mathbb{R}^m)$$, $$m \geqslant 5$$, Duke math. J., 47, 57-80, (1980) · Zbl 0437.47009 [9] Jensen, A., Spectral properties of Schrödinger operators and time-decay of the wave function. results in $$L^2(\mathbb{R}^4)$$, J. math. anal. appl., 101, 491-513, (1984) [10] Jensen, A.; Kato, T., Spectral properties of Schrödinger operators and time-decay of the wave functions, Duke math. J., 46, 583-611, (1979) · Zbl 0448.35080 [11] Jensen, A.; Nenciu, G., A unified approach to resolvent expansions at thresholds, Rev. math. phys., 13, 6, 717-754, (2001) · Zbl 1029.81067 [12] Klainerman, S., Remark on the asymptotic behavior of the klein – gordon equation in $$\mathbb{R}^{n + 1}$$, Comm. pure appl. math., 46, 2, 137-144, (1993) · Zbl 0805.35104 [13] Komech, A.; Kopylova, E., Scattering of solitons for Schrödinger equation coupled to a particle, Russ. J. math. phys., 13, 2, 158-187, (2006) · Zbl 1118.35040 [14] Komech, A.; Kopylova, E., Weighted energy decay for the 3D klein – gordon equation, J. differential equations, 248, 3, 501-520, (2010) · Zbl 1185.35023 [15] Komech, A.; Kopylova, E.; Kunze, M., Dispersive estimates for 1D discrete Schrödinger and klein – gordon equations, Appl. anal., 85, 12, 1487-1508, (2006) · Zbl 1121.39015 [16] Komech, A.; Kopylova, E.; Vainberg, B., On dispersive properties of discrete 2D Schrödinger and klein – gordon equations, J. funct. anal., 254, 2227-2254, (2008) · Zbl 1148.35087 [17] Kopylova, E., On dispersive decay for discrete 3D Schrödinger and klein – gordon equations, Algebra anal., 21, 5, 87-113, (2009) [18] Lundberg, L.-E., Spectral and scattering theory for the klein – gordon equation, Comm. math. phys., 31, 3, 243-257, (1973) · Zbl 1125.35391 [19] Marshall, B.; Strauss, W.; Wainger, S., $$L^p - L^q$$ estimates for the klein – gordon equation, J. math. pures appl., Sér. IX, 59, 417-440, (1980) · Zbl 0457.47040 [20] Murata, M., Asymptotic expansions in time for solutions of Schrödinger-type equations, J. funct. anal., 49, 10-56, (1982) · Zbl 0499.35019 [21] Nikiforov, A.F.; Uvarov, V.B., Special functions of mathematical physics; A unified introduction with applications, (1988), Birkhäuser Basel · Zbl 0694.33005 [22] Pego, R.L.; Weinstein, M.I., Asymptotic stability of solitary waves, Comm. math. phys., 164, 305-349, (1994) · Zbl 0805.35117 [23] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering for nonintegrable equations, Comm. math. phys., 133, 119-146, (1990) · Zbl 0721.35082 [24] Soffer, A.; Weinstein, M.I., Multichannel nonlinear scattering for nonintegrable equations II. the case of anisotropic potentials and data, J. differential equations, 98, 376-390, (1992) · Zbl 0795.35073 [25] Soffer, A.; Weinstein, M.I., Resonances, radiation damping and instability in Hamiltonian nonlinear wave equations, Invent. math., 136, 1, 9-74, (1999) · Zbl 0910.35107 [26] Schechter, M., The klein – gordon equation and scattering theory, Ann. physics, 101, 601-609, (1976) · Zbl 0347.35065 [27] Schlag, W., Dispersive estimates for Schrödinger operators, a survey, (), 255-285 · Zbl 1143.35001 [28] Vainberg, B.R., Behavior for large time of solutions of the klein – gordon equation, Trans. Moscow math. soc., 30, 139-158, (1976) · Zbl 0318.35051 [29] Vainberg, B.R., Asymptotic methods in equations of mathematical physics, (1989), Gordon and Breach New York · Zbl 0743.35001 [30] Weder, R.A., Scattering theory for the klein – gordon equation, J. funct. anal., 27, 100-117, (1978) · Zbl 0366.35030 [31] Weder, R.A., The $$L^p$$-$$L^{p^\prime}$$ estimate for the Schrödinger equation on the half-line, J. math. anal. appl., 281, 1, 233-243, (2003) · Zbl 1032.34081 [32] Yajima, K., The $$W^{k, p}$$-continuity of wave operators for Schrödinger operators., J. math. soc. Japan, 47, 3, 551-581, (1995) · Zbl 0837.35039
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.