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Large time behavior of solutions to semilinear systems of wave equations. (English) Zbl 1114.35129
The authors study asymptotic behavior of radially symmetric solutions of the initial value problem to the coupled wave equations \[ u_{tt}-c_1\triangle u=| v_t| ^p,\;v_{tt}-c_2\triangle v=| u_t| ^q, \;t>0, \;x\in \mathbb{R}^3. \] Under some conditions, they find small global solution of the problem and prove its convergence to a modified “free profile”. Moreover, they find some conditions, for which a global solution cannot exist.

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
35L15 Initial value problems for second-order hyperbolic equations
35L55 Higher-order hyperbolic systems
Full Text: DOI
[1] Alinhac, S.: An example of blowup at infinity for a quasilinear wave equation. Astérisque, Autour de l’analyse microlocale 284, 1–91 (2003) · Zbl 1053.35097
[2] Alinhac, S.: Semilinear hyperbolic systems with blowup at infinity. Preprint · Zbl 1122.35068
[3] Deng, K.: Blow-up of solutions of some nonlinear hyperbolic systems. Rocky Mountain J. Math. 29, 807–820 (1999) · Zbl 0945.35054
[4] Hidano, K., Tsutaya, K.: Global existence and asymptotic behavior of solutions for nonlinear wave equations. Indiana Univ. Math. J. 44, 1273–1305 (1995) · Zbl 0858.35085
[5] John, F.: Blow-up of solutions for quasi-linear wave equations in three space dimensions. Comm. Pure Appl. Math. 34, 29–51 (1981) · Zbl 0453.35060
[6] Kubo, H.: Asymptotic behaviors of solutions to semilinear wave equations with initial data of slow decay. Math. Methods in Appl. Sci. 17, 953–970 (1994) · Zbl 0807.35094
[7] Kubo, H., Kubota, K.: Asymptotic behavior of classical solutions to a system of semilinear wave equations in low space dimensions. J. Math. Soc. Japan. 53, 875–912 (2001) · Zbl 1016.35050
[8] Kubo, H., Kubota, K.: Existence and asymptotic behavior of radially symmetric solutions to a semilinear hyperbolic system in odd space dimensions. Chinese Ann. Math. Ser. B (to appear in) · Zbl 1129.35045
[9] Lindblad, H., Rodnianski, I.: Global existence for the Einstein vacuum equations in wave coordinates. Comm. Math. Phys. 256, 43–110 (2005) · Zbl 1081.83003
[10] Sideris, T.C.: Global behavior of solutions to nonlinear wave equations in three space dimensions. Comm. Partial Differential Equations 8, 1291–1323 (1983) · Zbl 0534.35069
[11] Strauss, W.A.: “Nonlinear wave equations”. CBMS Regional Conference Series in Mathematics, 73, American Math. Soc., Providence, RI, 1989 · Zbl 0714.35003
[12] Takamura, H.: Global existence for nonlinear wave equations with small data of noncompact support in three space dimensions. Comm. Partial Differential Equations 17, 189–204 (1992) · Zbl 0757.35046
[13] Tzvetkov, N.: Existence of global solutions to nonlinear massless Dirac system and wave equation with small data. Tsukuba J. Math. 22, 193–211 (1998) · Zbl 0945.35075
[14] Xu, W.: Blowup for systems of semilinear wave equations with small initial data. J. Partial Diff. Eqs. 17, 198–206 (2004) · Zbl 1066.35061
[15] Yokoyama, K.: Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions. J. Math. Soc. Japan. 52, 609–632 (2000) · Zbl 0968.35081
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