English

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

Analysis of PDEs 2026-03-24 v1

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 utuxxt+(f(u))x(f(u))xxx+(g(u)+f(u)2ux2)x+λ(uuxx)=0u_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 H1H^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.

Keywords

Cite

@article{arxiv.2603.20649,
  title  = {Global conservative weak solutions and global strong solutions for a class of weakly dissipative nonlinear dispersive wave equations},
  author = {Yiyao Lian and Zhenyu Wan and Zhaoyang Yin},
  journal= {arXiv preprint arXiv:2603.20649},
  year   = {2026}
}