Differential and Integral Equations

Nonlinear pseudodifferential equations on a segment

Elena I. Kaikina

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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 $\delta \in \left( 0,2-\alpha \right) .$

Article information

Source
Differential Integral Equations, Volume 18, Number 2 (2005), 195-224.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060229

Mathematical Reviews number (MathSciNet)
MR2106102

Zentralblatt MATH identifier
1212.35417

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10]
Secondary: 35B40: Asymptotic behavior of solutions 35S15: Boundary value problems for pseudodifferential operators

Citation

Kaikina, Elena I. Nonlinear pseudodifferential equations on a segment. Differential Integral Equations 18 (2005), no. 2, 195--224. https://projecteuclid.org/euclid.die/1356060229


Export citation