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September/October 2016 Initial boundary value problem of the Hamiltonian fifth-order KdV equation on a bounded domain
Bing-Yu Zhang, Deqin Zhou
Adv. Differential Equations 21(9/10): 977-1000 (September/October 2016).

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 $H^s(\mathbb R)$.

Citation

Download Citation

Bing-Yu Zhang. Deqin Zhou. "Initial boundary value problem of the Hamiltonian fifth-order KdV equation on a bounded domain." Adv. Differential Equations 21 (9/10) 977 - 1000, September/October 2016.

Information

Published: September/October 2016
First available in Project Euclid: 14 June 2016

zbMATH: 1375.35470
MathSciNet: MR3513123

Subjects:
Primary: 35B65 , 35Q53

Rights: Copyright © 2016 Khayyam Publishing, Inc.

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Vol.21 • No. 9/10 • September/October 2016
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