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

Abstract

We update here the results on the decay of solutions to a nonlocal water wave equation that reads $$\begin{aligned} u _t+u_x+\dfrac{1}{\sqrt{\pi }}\frac{\partial }{\partial t} \int _0^t\dfrac{u(s)}{\sqrt{t-s}} ds+ u u_x= u_{xx}, \end{aligned}$$where $$\displaystyle \dfrac{1}{\sqrt{\pi }}\dfrac{\partial }{\partial t} \int _0^t\dfrac{u(s)}{\sqrt{t-s}} ds $$ is the Riemann–Liouville half-order derivative.