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Potential well method for initial boundary value problem of the generalized double dispersion equations. (English) Zbl 1151.35003

The authors study the initial-value problem for the following (generalized) double dispersion equation, \[ u_{tt} - u_{xx} - u_{xxtt} + u_{xxxx} = f(u)_{xx}, \qquad x \in (0,l), \quad t > 0, \] with periodic boundary conditions \(u(0,t) = u(l,t) = u_{xx}(0,t) = u_{xx}(l,t) = 0\). Here \(f\) is a continuous function satisfying some technical assumptions; it can typically be thought of as \(f(u) = u^q\), for \(q > 1\).
The paper is similar to another contribution of the authors [J. Math. Anal. Appl. 338, No. 2, 1169–1187 (2008; Zbl 1140.35011)]. For other similar results, see also the investigation by S. Wang and G. Chen [Nonlinear Anal., Theory Methods Appl. 64, No. 1 (A), 159–173 (2006; Zbl 1092.35056)].

MSC:

35A25 Other special methods applied to PDEs
35L82 Pseudohyperbolic equations
35Q53 KdV equations (Korteweg-de Vries equations)
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