Blowup issues for a class of nonlinear dispersive wave equations

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 $x\to-\infty$

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.Comment: Corrected proofs. To appear on J. Funct. Ana

Smoothness of the flow map for low-regularity solutions of the Camassa-Holm equations

It was recently proven by De Lellis, Kappeler, and Topalov that the periodic Cauchy problem for the Camassa-Holm equations is locally well-posed in the space Lip (T) endowed with the topology of H^1 (T). We prove here that the Lagrangian flow of these solutions are analytic with respect to time and smooth with respect to the initial data. These results can be adapted to the higher-order Camassa-Holm equations describing the exponential curves of the manifold of orientation preserving diffeomorphisms of T using the Riemannian structure induced by the Sobolev inner product H^l (T), for l in N, l > 1 (the classical Camassa-Holm equation corresponds to the case l=1): the periodic Cauchy problem is locally well-posed in the space W^{2l-1,infty} (T) endowed with the topology of H^{2l-1} (T) and the Lagrangian flows of these solutions are analytic with respect to time with values in W^{2l-1,infty} (T) and smooth with respect to the initial data. These results extend some earlier results which dealt with more regular solutions. In particular our results cover the case of peakons, up to the first collision

Global existence and propagation speed for a generalized Camassa-Holm model with both dissipation and dispersion

In this paper, we study a generalized Camassa–Holm (gCH) model with both dissipation and dispersion, which has (N+1)-order nonlinearities and includes the following three integrable equations: the Camassa–Holm, the Degasperis–Procesi, and the Novikov equations, as its reductions. We first present the local well-posedness and a precise blow-up scenario of the Cauchy problem for the gCH equation. Then, we provide several sufficient conditions that guarantee the global existence of the strong solutions to the gCH equation. Finally, we investigate the propagation speed for the gCH equation when the initial data are compactly supported

Wave Breaking for the Modified Two-Component Camassa-Holm System

Some new sufficient conditions to guarantee wave breaking for the modified two-component Camassa-Holm system are established

On the Cauchy Problem for the b

In this paper, we consider b-family equations with a strong dispersive term. First, we present a criterion on blow-up. Then global existence and persistence property of the solution are also established. Finally, we discuss infinite propagation speed of this equation

Rogue peakon, well-posedness, ill-posedness and blow-up phenomenon for an integrable Camassa-Holm type equation

In this paper, we study an integrable Camassa-Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution - called {\sf rogue peakon}, that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space $B_{p,r}^{s}$ with $1\leq p,r\leq\infty$, $s>\max \left\{1+1/p,3/2\right\}$ or $B_{2,1}^{3/2}$, and then prove the ill-posedness of the solution in $B_{2,\infty}^{3/2}$. Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if $m_0(x)=u_0-u_{0xx}\geq(\not\equiv) 0$, then the corresponding solution exists globally, meanwhile, if $m_0(x)\leq(\not\equiv) 0$, then the corresponding solution blows up in a finite time.Comment: 23 pages, 6 figure