Global behavior for Whitham type equations on a finite interval (Q948956)

The author deals with a pseudodifferential equations on a finite interval, considering the case of large initial data. The purpose of this paper is to prove a global existence of solutions to the initial-boundary value problem of the form \[ \begin{cases} u_t+\lambda| u |^\sigma u+\mathbb{K} u=0, & t>0,\;x\in(0,a)\\ u(0,x)=u_0(x), & x\in [0,a]\\ u(a,t)=0, & t>0,\end{cases}\tag{1} \] where \(\lambda>0\) and \(\mathbb{K}\) is a pseudodifferential operator defined by the inverse Laplace transform. Moreover, the author is interested in to find the main term of the asymptotic representations of solutions.