# Wave breaking for shallow water models with time decaying solutions

## TL;DR

This paper studies a family of Camassa-Holm type equations with linear and nonlinear terms, establishing well-posedness, conserved quantities, and conditions for wave breaking, especially when the linear coefficient is small.

## Contribution

It introduces a new analysis of wave breaking for a generalized Camassa-Holm type equation with time-decaying solutions and linear perturbations.

## Key findings

- Energy functional decreases monotonically over time.
- Wave breaking occurs under specific conditions, notably small linear coefficient.
- Conserved quantities are identified and used to analyze solution behavior.

## 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.