Abstract A family of Camassa-Holm type equations with a linear term and cubic and quartic nonlinearities is considered. Local well-posedness results are established via Kato's approach. Conserved quantities for the equation are determined and from them we prove that the energy functional of the solutions is a time-dependent, monotonically decreasing function of time, and bounded from above by the Sobolev norm of the initial data under some conditions. The existence of wave breaking phenomenon is investigated and necessary conditions for its existence are obtained. In our framework the wave breaking is guaranteed, among other conditions, when the coefficient of the linear term is sufficiently small, which allows us to interpret the equation as a linear perturbation of some recent Camassa-Holm type equations considered in the literature.

中文翻译：

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具有时间衰减解的浅水模型的波浪破碎

摘要 考虑了具有线性项和三次和四次非线性的Camassa-Holm 型方程族。局部适定结果是通过 Kato 的方法建立的。方程的守恒量被确定，我们从中证明了解的能量泛函是时间相关的、单调递减的时间函数，并且在某些条件下由初始数据的 Sobolev 范数从上方限定。考察了破浪现象的存在，得到了破浪现象存在的必要条件。在我们的框架中，除其他条件外，当线性项的系数足够小时，波浪破碎得到保证，这使我们能够将方程解释为文献中考虑的一些最近的 Camassa-Holm 类型方程的线性扰动。