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Global existence and asymptotic behaviour of solutions for the generalized Boussinesq equation. (English) Zbl 1161.35470
Summary: We prove the existence and the uniqueness of global solution for the Cauchy problem for the generalized Boussinesq equation. Under some assumptions, we also show that the $L_\infty $ norm of small solution of the Cauchy problem for the generalized Boussinesq equation decays to zero as $t$ tends to infinity.

35Q35PDEs in connection with fluid mechanics
35B45A priori estimates for solutions of PDE
35B40Asymptotic behavior of solutions of PDE
76B15Water waves, gravity waves; dispersion and scattering, nonlinear interaction
Full Text: DOI
[1] Akmel, D. G.: Global existence and decay for solution to the bad Boussinesq equation in two space dimensions. Applicable analysis 83, No. 1, 17-36 (2004) · Zbl 1049.35046
[2] Boussinesq, M. J.: Essai\dot{} sur la théorie des eaux courantes, mémoires présentés par divers savants á I. Académie des sciences inst. France, séries 2, No. 3, 1-680 (1877)
[3] Clarkson, P.: New exact solution of the Boussinesq equation. European journal of applied mathematics 1, 279-300 (1990) · Zbl 0721.35074
[4] Hrusa, W. J.: Global existence and asymptotic stability for a semilinear hyperbolic Volterra equation with large initial data. SIAM journal on mathematical analysis 16, 110-134 (1985) · Zbl 0571.45007
[5] Kalantarov, V. K.; Ladyzhenskaya, O. A.: The occurrence of collapse for quasilinear equation of parabolic and hyperbolic types. Journal of soviet mathematics 10, 53-70 (1978) · Zbl 0388.35039
[6] Levine, H. A.; Sleeman, B. O.: A note on the non-existence of global solutions of initial boundary value problems for the Boussinesq equation utt=uxx+3uxxxx-$12(u2)$xx. Journal of mathematical analysis and applications 107, 206-210 (1985) · Zbl 0591.35010
[7] Linares, F.; Scialom, M.: Asymptotic behavior of solutions of a generalized Boussinesq type equation. Nonlinear analysis 25, 1147-1158 (1995) · Zbl 0847.35109
[8] Linares, F.: Global existence of small solutions for a generalized Boussinesq equation. Journal of differential equations 106, 257-293 (1993) · Zbl 0801.35111
[9] Liu, Y.: Existence and blow up of a nonlinear pochhammercchree equation. Indiana university mathematics journal 45, No. 3, 797-816 (1996) · Zbl 0883.35116
[10] Liu, Y.: Decay and scattering of small solutions of a generalized Boussinesq equation. Journal of functional analysis 147, 51-68 (1997) · Zbl 0884.35129
[11] Li, T. T.; Chen, Y. M.: Global classical solution for nonlinear evolution equations. Pitman monographs and surveys in pure and applied mathematics 45 (1992)
[12] Schneider, G.; Eugene, C. W.: Kawahara dynamics in dispersive media. Physica D 152--153, 108-110 (2001)
[13] E.M. Stein, Singular integrals and differentiability properties of function, Princeton University, Princeton, NJ, 1970 · Zbl 0207.13501
[14] Tayler, M. E.: Partial differential equations III. Nonlinear equations (1996)
[15] Wang, S. B.; Chen, G. W.: Small amplitude solutions of the generalized imbq equation. Journal of mathematical analysis and applications 264, 846-866 (2002) · Zbl 1136.35425
[16] Wang, S. B.; Chen, G. W.: Cauchy problem of the generalized double dispersion equation. Nonlinear analysis theory, methods and applications 64, 159-173 (2006) · Zbl 1092.35056
[17] Wang, Y.; Mu, C. L.: Global existence and blow-up of the solutions for the multidimensional generalized Boussinesq equation. Mathematical methods in the applied sciences 30, 1403-1417 (2007) · Zbl 1127.35052
[18] Wang, Y.; Mu, C. L.: Blow-up and scattering of solution for a generalized Boussinesq equation. Applied mathematics and computation 188, 1131-1141 (2007) · Zbl 1118.35039