Global existence of smooth solutions and stability of solitary waves for a generalized Boussinesq equation. (English) Zbl 0654.35018

Summary: Certain generalizations of one of the classical Boussinesq-type equations, \[ u_{tt}=u_{xx}-(u\quad 2+u_{xx})_{xx},\quad (*) \] are considered. It is shown that the initial-value problem for this type of equation is always locally well posed. It is also determined that the special, solitary-wave solutions of these equations are nonlinearly stable for a range of their phase speeds. These two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations. This contrasts with the results of blow up obtained recently by Kalantarov and Ladyzhenskaya for the same type of equation, and casts additional light upon the results for the original version (*) of this class of equations obtained via inverse-scattering theory by Deift, Tomei and Trubowitz.


35G25 Initial value problems for nonlinear higher-order PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
35B40 Asymptotic behavior of solutions to PDEs
35B35 Stability in context of PDEs
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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