Abstract. We prove that the only global strong solution of the periodic rod equation vanishing in at least one point (t0, x0) ∈ R + × 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. 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.

Abstract. In this paper we consider the nonlinear dispersive wave equation on the realu 2 x x = 0, that for appropriate choices of the functions f and g includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa-Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (i.e., a criterion involving only the properties of the data u0 in a neighbourhood of a single point) simplifying and extending earlier blowup criteria for this equation. Our arguments apply both to the finite and infinite energy case, yielding the finite time blowup of strong solutions with possibly different behavior as x → +∞ and x → −∞.