Advances in Differential Equations

Initial boundary value problem of the Hamiltonian fifth-order KdV equation on a bounded domain

Bing-Yu Zhang and Deqin Zhou

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

In this paper, we consider the initial boundary value problem (IBVP) of the Hamiltonian fifth-order KdV equation posed on a finite interval $(0,L)$, $$ \begin{cases} \partial_t u-\partial_x^{5}u= c_1 u\partial_x u+ c_2 u^2 \partial _x u + 2b\partial_x u\partial_x^{2}u+bu\partial_x^{3} u, \quad x\in (0,L), \ t>0 \\ u(0,x)=\phi (x) , \ x\in (0,L)\\ u(t,0)=\partial_x u(t,0)=u(t,L)=\partial_x u(t,L)=\partial_x^{2}u(t,L)=0, \quad t>0, \end{cases} $$ and show that, given $0\leq s\leq 5$ and $T>0$, for any $\phi \in H^s (0,L) $ satisfying the natural compatibility conditions, the IBVP admits a unique solution $$ u\in L^{\infty}_{loc} (\mathbb R^+; H^s(0,L))\cap L^2 _{loc}(\mathbb R^+; H^{s+2} (0,L)). $$ Moreover, the corresponding solution map is shown to be locally Lipschtiz continuous from $L^2 (0,L)$ to $L^{\infty}(0,T; L^2 (0,L))\cap L^2 (0,T; H^2 (0,L))$ and from $H^5 (0,L)$ to $L^{\infty}(0,T; H^5 (0,L))\cap L^2 (0,T; H^7 (0,L))$, respectively, for any given $T>0$ . This is in sharp contrast to the pure initial value problem (IVP) of the equation posed on the whole line $\mathbb R$, $$ \begin{cases} \partial_t v-\partial_x^{5}v= c_1 v\partial_x v+ c_2 v^2 \partial _x v + 2b\partial_x v\partial_x^{2}u+bv\partial_x^{3} v, \quad x\in \mathbb R, \ t\in \mathbb R \\ v(0,x)=\psi (x) , \ x\in \mathbb R, \end{cases} $$ which is known to be (globally) well-posed in the space $H^s (\mathbb R)$ for $s\geq 2$ and the corresponding solution map is continuous, but fails to be uniformly continuous on any ball in $H^s(\mathbb R)$.

Article information

Source
Adv. Differential Equations Volume 21, Number 9/10 (2016), 977-1000.

Dates
First available in Project Euclid: 14 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ade/1465912588

Mathematical Reviews number (MathSciNet)
MR3513123

Subjects
Primary: 35Q53: KdV-like equations (Korteweg-de Vries) [See also 37K10] 35B65: Smoothness and regularity of solutions

Citation

Zhou, Deqin; Zhang, Bing-Yu. Initial boundary value problem of the Hamiltonian fifth-order KdV equation on a bounded domain. Adv. Differential Equations 21 (2016), no. 9/10, 977--1000. https://projecteuclid.org/euclid.ade/1465912588.


Export citation