Blowup issues for a class of nonlinear dispersive wave equations

Lorenzo Brandolese (1) , Manuel Fernando Cortez (1)

1 EDPA - Équations aux dérivées partielles, analyse

In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, 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 $u_0$ 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 $x\to+\infty$ and $x\to-\infty$.

Mots clés: Camassa--Holm; shallow water; Wave breaking; Rod equation

Domaines: Equations aux dérivées partielles [math.AP]

Lorenzo Brandolese, Manuel Fernando Cortez. Blowup issues for a class of nonlinear dispersive wave equations. Journal of Differential Equations, 2014, 256 (12), pp.3981-3998. ⟨hal-00957008⟩