Large time asymptotics for the Ott-Sudan-Ostrovskiy type equations on a segment

Abstract

We study the initial-boundary value problems for the nonlinear nonlocal equation on a segment $\left( 0,a\right) $ \begin{equation} \left\{ \begin{array}{c} u_{t}+\lambda \left\vert u\right\vert \text{ }u+C_{1}\int_{0}^{x}\frac{ u_{ss}(s,t)}{\sqrt{x-s}}ds=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=h_{1}(t),u_{x}(0,t)=h_{2}(t),t>0, \end{array} \right. \label{2} \end{equation} where $\lambda \in \mathbf{R}$ and the constant $C_{1}$ is chosen by the condition of the dissipation, such that $ {\rm Re\,}C_{1}p^{\frac{3}{2}}>0$ for ${\rm Re\,}p=0.$ The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem and to find the main term of the asymptotic representation of solutions.