On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point $(t_0,x_0)$ is the identically zero solution. Such conclusion holds provided the physical parameter $\gamma$ of the model (related to the finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa--Holm equation, corresponding to $\gamma=1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincar\'e inequalities.