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

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 $u_t-u_{xxt}+(f\left(u\right){)}_x-(f\left(u\right){)}_{xxx}+{\left(g\left(u\right)+\frac{f^{\prime \prime }\left(u\right)}{2}{u}_x^2\right)}_x+\hat{I}»\left(u-u_{xx}\right)=0$. 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.