Nonlinear nonlocal Ott-Sudan-Ostrovskiy type equationson a segment

Abstract

We study the global existence and large time asymptotic behavior of solutions to the initial-boundary value problems for the nolinear nonlocal equation on a segment $\left( 0,a\right) $ \begin{equation} \left\{ \begin{array}{c} u_{t}+u\text{ }u_{x}+\frac{C_{1}}{\pi }\partial _{x}\int_{0}^{x}\frac{% u_{s}(s,t)}{\sqrt{x-s}}ds=0,\text{ }t>0, \\ u(x,0)=u_{0}(x), \\ u(a,t)=0,t>0% \end{array}% \right. \label{1} \end{equation} and \begin{equation} \left\{ \begin{array}{c} u_{t}+u\text{ }u_{x}+\frac{C_{1}}{\pi }\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)=u_{x}(0,t)=0,t>0% \end{array} \right. \label{2} \end{equation} where the constant $C_{1}$ is choosen by a dissipative condition, such that ${\mathrm Re} C_{1}p^{\frac{3}{2}}>0$ for ${\mathrm Re}p=0.$ We prove that if the initial data $u_{0}\in \mathbf{L}^{\infty }\left( 0,a\right) $ is small enough$,$ then there exists a unique solution of problems (0.1) and (0.2) $u\in \mathbf{C}([0,+\infty );\mathbf{L}^{2}\left( 0,a\right) ) $ $\cap \mathbf{C}(\mathbf{R}^{+};\mathbf{H}^{1}\left( 0,a\right) ).$ Moreover there exists a constant $A$ such that the solution has the following large time asymptotics uniformly with respect to $x\in \left( 0,a\right) $ \begin{equation*} u(x,t)=At^{-\frac{2}{3}}\Lambda \left( xt^{-\frac{2}{3}}\right) +O\left( t^{-% \frac{2(1+\delta )}{3}}\right) , \end{equation*}% where $\delta \in \left( 0,\frac{2}{3}\right) $ and \begin{equation*} \Lambda (s)=\frac{e^{-i\frac{\pi }{4}}\sqrt{2}}{2\pi i}\int_{0}^{+i\infty }e^{sz-C_{1}z^{\frac{3}{2}}}dz,\text{ }s>0. \end{equation*}