Global conservative weak solutions and global strong solutions for a class of weakly dissipative nonlinear dispersive wave equations

TL;DR

This paper establishes the global existence of weak and strong solutions for a class of weakly dissipative nonlinear dispersive wave equations, including models like the Camassa-Holm equation, extending previous results.

Contribution

It provides new global existence results for both energy conservative weak solutions and strong solutions under various initial conditions for a broad class of equations.

Findings

Global existence of energy conservative weak solutions in weighted $H^1$ space.

Global strong solutions for small initial data.

Global strong solutions for sign-changing initial data.

Abstract

In this paper, we study the global existence of solutions of the Cauchy problem for a class of weakly dissipative nonlinear dispersive wave equations $u_t-u_{xxt}+(f\left(u\right))_x-(f\left(u\right))_{xxx}+\left(g\left(u\right)+\frac{f^{\prime\prime}\left(u\right)}{2}u_x^2\right)_x+\lambda\left(u-u_{xx}\right)=0$. This includes the weakly dissipative Camassa-Holm equation and the weakly dissipative hyperelastic rod wave equation as special cases. Specifically, we establish three global existence results: one concerning the energy conservative weak solutions in a time-weighted $H^1$ space, and the other two concerning strong solutions, which include the cases of small initial data and sign-changing initial data. Our results recover and extend many known results for several classical models.