Global Weak Solutions to a Generalized Hyperelastic-rod Wave Equation

Abstract

We consider a generalized hyperelastic-rod wave equation (or generalized Camassa--Holm equation) describing nonlinear dispersive waves in compressible hyperelastic rods. We establish existence of a strongly continuous semigroup of global weak solutions for any initial data from $H^1(\R)$. We also present a "weak equals strong" uniqueness result.