Potential well method for initial boundary value problem of the generalized double dispersion equations (Q928114)

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 \textit{S. Wang} and \textit{G. Chen} [Nonlinear Anal., Theory Methods Appl. 64, No. 1 (A), 159--173 (2006; Zbl 1092.35056)].