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Initial boundary value problem of the Hamiltonian fifth-order KdV equation on a bounded domain. (English) Zbl 1375.35470

Summary: We consider the initial boundary value problem (IBVP) of the Hamiltonian fifth-order KdV equation posed on a finite interval \((0, L)\), \[ \begin{cases} \partial_tu - \partial^5_xu = c_1u\partial_xu + c_2u^2 \partial_xu + 2b\partial_xu\partial^2_xu + bu\partial^3_xu,\quad x \in (0, L),\;t > 0\\ u(0, x) = \phi (x),\;x \in (0, L)\\ u(t, 0) = \partial_xu(t, 0) = u(t, L) = \partial_xu(t, L) = \partial^2_xu(t, L) = 0,\quad t > 0, \end{cases} \] 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 \(\mathbb R\), \[ \begin{cases} \partial_tv - \partial^5_xv = c_1v\partial_xv + c_2v^2 \partial_xv + 2b\partial_xv\partial^2_xu + bv\partial^3_xv,\quad x \in \mathbb R,\;t \in \mathbb R\\ v(0, x) = \psi (x),\;x \in \mathbb R, \end{cases} \] 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 \(H^s (\mathbb R)\).

MSC:

35Q53 KdV equations (Korteweg-de Vries equations)
35B65 Smoothness and regularity of solutions to PDEs
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Full Text: Euclid