Blowup issues for a class of nonlinear dispersive wave equations

Abstract

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 → −∞.

On permanent and breaking waves in hyperelastic rods and rings

“…As a comparison, the blowup criterion in [3] for solutions in H s (R) of the Camassa-Holm equation reads inf R (u 0 + |u 0 |) < 0; on the other hand, according to (1.5), in the periodic case the condition inf S (u 0 + 0.515 |u 0 |) < 0 would be enough for the development of a singularity. In general, for γ ∈ [1,4], the coefficient β γ in (1.4) is considerably lower than the corresponding coefficient β γ,R computed in [3] for the blowup criterion of non-periodic solutions.…”

“…Now integrating these inequalities on (t, T * ) we get the blowup rate (2.5) with x(t) = q(t, x 0 ). 4. The minimization problem in the limit case β = e+1 e−1 and in the case β = 1 4.1.…”

Blow‐up for the b‐family of equations

“…The Eq. is physically relevant as it also describes the nonlinear dispersive waves in compressible hyperelastic rods . It is convenient to rewrite the Cauchy problem associated with the dispersionless case of in the following weak form: where p ( x ) is the fundamental solution of the operator in .…”

“…The Eq. C-H is physically relevant as it also describes the nonlinear dispersive waves in compressible hyperelastic rods [7,9,10]. It is convenient to rewrite the Cauchy problem associated with the dispersionless case of (C-H) in the following weak form:…”

On the Dynamics of Zero-Speed Solutions for Camassa–Holm-Type Equations

“…These are the so called local-in-space blow-up criteria, which involve only properties of the initial data in a small neighborhood of a single point. In that sense, such criteria are more general, see [17,18,19] for further details. Recall that these results do not exclude the existence of global conservative and dissipative weak solutions, as explained above by [21,22].…”