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Global existence for the Benjamin-Bona-Mahony equation in arbitrary dimensions. (English) Zbl 0591.35012
The Benjamin-Bona-Mahony equation \[ u_ t-u_{xxt}+uu_ x=0,\quad x\in\mathbb R,\quad t\geq 0, \] models long waves in a nonlinear dispersive system. Global existence and uniqueness results for this equation were established by T. B. Benjamin, J. L. Bona and J. J. Mahony [Philos. Trans. R. Soc. Lond., A 272, 47–78 (1972; Zbl 0229.35013)]. In higher dimensions, the generalized BBM equation is \[ u_ t-\Delta_ xu_ t+\mathop{div}(\phi u)=0,\quad x\in\mathbb R^ d,\quad t\geq 0. \] In two papers by B. Calvert [Math. Proc. Camb. Philos. Soc. 79, 545–561 (1976; Zbl 0319.47030) and B. J. Wichnoski and the second author [Nonlinear Anal., Theory Methods Appl. 4, 665–675 (1980; Zbl 0447.35068)], existence results were obtained in dimension \(d\leq 5\) when \(\phi'\) satisfies a polynomial-like growth bound. Here the results of Wichnoski and the second author are extended to all dimensions.
Reviewer: G. Gudmundsdottir

MSC:
35G25 Initial value problems for nonlinear higher-order PDEs
35Q53 KdV equations (Korteweg-de Vries equations)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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References:
[1] Adams, R.A., Sobolev spaces, (1975), Academic Press New York · Zbl 0186.19101
[2] Benjamin, T.B.; Bona, J.L.; Mahony, J.J., Model equations for long waves in nonlinear dispersive systems, Phil. trans. R. soc. lond., A272, 47-78, (1972) · Zbl 0229.35013
[3] Calvert, B., The equation A(t, u(t))’ + B(t, u(t)) = 0, Math. proc. camb. phil. soc., 79, 545-561, (1976) · Zbl 0319.47030
[4] Friedman, A., Partial differential equations, (1969), Holt, Rinehart & Winston New York
[5] Goldstein, J.A.; Wichnoski, B., On the Benjamin-Bona-Mahony equation in higher dimensions, Nonlinear analysis, 4, 665-675, (1980) · Zbl 0447.35068
[6] Witham, G.G., Linear and nonlinear waves, (1974), John Wiley New York
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