On an initial-boundary value problem for the hyperelastic rod wave equation. (English) Zbl 1252.35131

The authors consider a third-order dispersive wave equation \[ u_t - u_{txx} + 3 u u_x = \gamma (2 u_x u_{xx} + u u_{xxx}), \] on the spatial interval \([0,1]\) equipped with inhomogeneous boundary conditions for \(u\) and \(u_x\) at \(x = 0\) and \(x = 1\). The initial data is in \(H^1(0,1)\). When \(\gamma = 1\), the dispersive wave equation coincides with the Camassa-Holm equation. In a general setting, it describes small amplitude, finite length radial deformation waves in cylindrical compressible hyperelastic rods. The constant \(\gamma\) is given in terms of the material constants and the prestress of the rod.
The purpose of this paper is to establish the existence result for a global-in-time weak solutions of the initial-boundary value problem. This is proven by passing to the limit in a sequence of approximate solutions. In addition, the authors provide a uniqueness result, based on the principle of “weak equals strong” solutions.


35G31 Initial-boundary value problems for nonlinear higher-order PDEs
74K10 Rods (beams, columns, shafts, arches, rings, etc.)