A modified Water Wave model with a non local viscous dispersive term

Abstract

In this paper, we consider the modified water wave model with a nonlocal viscous term
\[
 u_t+u_x- u_{xx}+\dfrac{1}{\sqrt \pi}\frac{\partial}{\partial t} \int_0^t\dfrac{u(s)}{\sqrt{t-s}} ds=- u^2 u_x,
\]
where $\frac{1}{\sqrt \pi}\frac{\partial}{\partial t}  \int_0^t\frac{u(s)}{\sqrt{t-s}} ds$ is the Riemann-Liouville half derivative. We discuss theoretically and numerically the well-posedness of the model and the asymptotical behavior of the solutions.