Decay of solutions to a water wave model with a nonlocal viscous dispersive term

Abstract

In this article, we investigate a water wave model with a nonlocal viscous term \begin{equation*} u_t+u_x+\beta u_{xxx}+\frac{\sqrt{\nu}}{\sqrt{\pi}}\int_0^t \frac{u_t(s)}{\sqrt{t-s}}ds+uu_x=\nu u_{xx}. \end{equation*} The wellposedness of the equation and the decay rate of solutions are investigated theoretically and numerically