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Global small amplitude solutions for two-dimensional nonlinear Klein-Gordon systems in the presence of mass resonance. (English) Zbl 1239.35022
From the abstract: We consider a nonlinear system of two-dimensional Klein-Gordon equations with masses $m_1, m_2$ satisfying the resonance relation $m_2 = 2m_1 > 0$. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate $O(|t|-1)$ as $t \to\pm\infty$ in $L^\infty$. In particular, our new condition includes the Yukawa type interaction, which has been excluded from the null condition in the sense of {\it J.-M. Delort, D. Fang} and {\it R. Xue} [J. Funct. Anal. 211, No. 2, 288--323 (2004; Zbl 1061.35089)]. The aim in this paper is to give a new sufficient condition on the nonlinearities. Under this condition, we will show that the solution exists globally in time and it enjoys time decay property even in the resonant case.

35B40Asymptotic behavior of solutions of PDE
35L52Second-order hyperbolic systems, initial value problems
35L71Semilinear second-order hyperbolic equations
35Q40PDEs in connection with quantum mechanics
Full Text: DOI arXiv
[1] Delort, J. -M.: Existence globale et comportement asymptotique pour l’équation de Klein-Gordon quasi linéaire à données petites en dimension 1, Ann. sci. Éc. norm. Super. (4) 34, 1-61 (2001) · Zbl 0990.35119 · doi:10.1016/S0012-9593(00)01059-4 · numdam:ASENS_2001_4_34_1_1_0
[2] Delort, J. -M.; Fang, D.; Xue, R.: Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. funct. Anal. 211, 288-323 (2004) · Zbl 1061.35089 · doi:10.1016/j.jfa.2004.01.008
[3] Hayashi, N.; Naumkin, P. I.; Wibowo, Ratno Bagus Edy: Nonlinear scattering for a system of nonlinear Klein-Gordon equations, J. math. Phys. 49, 103501 (2008) · Zbl 1152.81467 · doi:10.1063/1.2990493
[4] S. Katayama, T. Ozawa, H. Sunagawa, A note on the null condition for quadratic nonlinear Klein-Gordon systems in two space dimensions, preprint, 2011. · Zbl 1244.35113
[5] Sunagawa, H.: On global small amplitude solutions to systems of cubic nonlinear Klein-Gordon equations with different mass terms in one space dimension, J. differential equations 192, 308-325 (2003) · Zbl 1028.35128 · doi:10.1016/S0022-0396(03)00125-6
[6] Sunagawa, H.: A note on the large time asymptotics for a system of Klein-Gordon equations, Hokkaido math. J. 33, 457-472 (2004) · Zbl 1065.35177
[7] Sunagawa, H.: Large time asymptotics of solutions to nonlinear Klein-Gordon systems, Osaka J. Math. 42, 65-83 (2005) · Zbl 1211.35196
[8] Sunagawa, H.: Remarks on the asymptotic behavior of the cubic nonlinear Klein-Gordon equations in one space dimension, Differential integral equations 18, 481-494 (2005) · Zbl 1212.35318
[9] Sunagawa, H.: Large time behavior of solutions to the Klein-Gordon equation with nonlinear dissipative terms, J. math. Soc. Japan 58, 379-400 (2006) · Zbl 1107.35087 · doi:10.2969/jmsj/1149166781
[10] Taflin, E.: Simple non-linear Klein-Gordon equations in two space dimensions, with long-range scattering, Lett. math. Phys. 79, 175-192 (2007) · Zbl 1123.35081 · doi:10.1007/s11005-006-0128-9
[11] Tsutsumi, Y.: Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. ekvac. 46, 41-62 (2003) · Zbl 1151.58303 · doi:10.1619/fesi.46.41