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 (_t0,_x0)∈R⁺×S¹$(t_0,x_0)\in {\mathbb{R}}^{+}\times {\mathbb{S}}^1$ is the identically zero solution. Such conclusion holds provided the physical parameter γ   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 γ=1$\gamma =1$. 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 Poincaré inequalities.