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Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation. (English) Zbl 1195.35072

This paper investigates the Cauchy problem on the real line for the weakly dissipative Camassa-Holm (wdCH) equation

${u}_{t}+3u{u}_{x}-{u}_{xxt}+\lambda \left(u-{u}_{xx}\right)=2{u}_{x}{u}_{xx}+u{u}_{xxx},\phantom{\rule{2.em}{0ex}}\left(1\right)$

where $\lambda >0$ is a constant. The non-dissipative limiting case $\lambda \to 0$ of (1) recovers the celebrated Camassa-Holm (CH) equation. As is noted therein, the wdCH equation admits solutions whose properties are markedly different to those of the CH equation. Significantly, and as a consequence of the extra dissipative term, (1) has no travelling-wave solutions.

The main achievement of this work is to obtain a global existence result for (strong) solutions of the wCH equation having certain profiles. The authors demonstrate that, following an approach due to A. Constantin [Ann. Inst. Fourier 50, No. 2, 321–362 (2000; Zbl 0944.35062)], solutions of (1) either exist globally in time or blow up in finite time for a wide class of initial profiles.

The work begins by recapping some known results on the well-posedness and blow-up scenarios for the wdCH equation. By first deriving a useful ${L}^{\infty }$-norm for strong solutions of (1), the authors proceed to establish a novel global existence theorem for these solutions. To conclude the study, they also prove a new blow-up result (for strong solutions) and determine a breaking point where the gradient of the solution becomes infinite. The results of this study extend and improve on those previously reported for the wdCH equation.

This study will be of interest to those working on nonlinear wave equations – especially shallow-water wave equations like the CH equation – and who are concerned with the analytic and dynamic character of their solutions.

##### MSC:
 35B44 Blow-up (PDE) 35G25 Initial value problems for nonlinear higher-order PDE 35Q35 PDEs in connection with fluid mechanics
##### Keywords:
solutions with profiles; strong solutions