Global existence of solutions to nonlinear dispersive wave equations

Abstract

We study the global existence of solutions to nonlinear dispersive wave equations \begin{equation*} \partial _{t}^{2}u+\frac{1}{\rho ^{2}}\left\vert \partial _{x}\right\vert ^{2\rho }u=\lambda \left\vert \partial _{t}u\right\vert ^{p-1}\partial _{t}u \end{equation*} in one space dimension, where $0<\rho \leq 2,\rho \neq 1,p>3$ and $\lambda \in \mathbf{C}.$