Nonlinear pseudodifferential equations on a segment

Abstract

We study the global existence and large-time asymptotic behavior of solutions to the initial/boundary-value problem for the nonlinear nonlocal Whitham equation on a segment $\left( 0,a\right), $ \begin{equation} \left\{ \begin{array}{c} u_{t}+uu_{x}+\mathbb{K}u=0,\text{ }t>0,x\in \left( 0,a\right) \\ u(x,0)=u_{0}(x),\text{ }x\in \left( 0,a\right) , \end{array} \right. \tag*{(0.1)} \end{equation} where the pseudodifferential operator $\mathbb{K}u$ on a segment $\left[ 0,a \right] $ is defined by \begin{align} \mathbb{K}u = \theta _{a}(x)\frac{1}{2\pi i}\int_{-i\infty }^{i\infty }e^{px}K(p) \Big ( \widehat{u}(p,t)-\frac{u(0,t)-e^{-pa}u(a,t)}{p}\Big) dp, \tag*{(0.2)} \end{align} where $K(p)=C_{\alpha }p^{\alpha },$ $\alpha \in ( \frac{3}{2},2 ) ,$ and $C_{\alpha }$ is chosen by the dissipation conditions. We prove that if the initial data $u_{0}\in \mathbf{L}^{\infty }(0,a)$ have a small norm $ \left\Vert u_{0}\right\Vert _{\mathbf{L}^{\infty }} < \varepsilon ,$ then there exists a unique solution $u\in \mathbf{C}\left( \left[ 0,\infty \right) ;\mathbf{L}^{2}(0,a)\right) $ $\cap \mathbf{C}\left( \left( 0,\infty \right) ;\mathbf{H}^{1}(0,a)\right) $ to problem (0.2). Moreover, there exists a function $A\left( x\right) \in \mathbf{L}^{\infty }(0,a)$ such that the solution has the following asymptotics for large time $ t\rightarrow \infty $: \begin{equation*} u(x,t)=A\left( x\right) Bt^{-\frac{1}{\alpha }}+O ( t^{-\frac{1+\delta }{ \alpha }} ) , \end{equation*} uniformly with respect to $x\in \left( 0,a\right) ,$ where $\delta \in \left( 0,2-\alpha \right) .$