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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 ${m}_{2}=2{m}_{1}>0$. We introduce a structural condition on the nonlinearities under which the solution exists globally in time and decays at the rate $O\left(|t|-1\right)$ as $t\to ±\infty$ 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 J.-M. Delort, D. Fang and 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.

##### MSC:
 35B40 Asymptotic behavior of solutions of PDE 35L52 Second-order hyperbolic systems, initial value problems 35L71 Semilinear second-order hyperbolic equations 35Q40 PDEs in connection with quantum mechanics
##### References:
 [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 · doi: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. [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