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An initial-boundary value problem for the Korteweg-de Vries equation posed on a finite interval. (English) Zbl 1022.35055
The authors of this interesting paper study the Korteweg-de-Vries (KdV) equation \(u_t+uu_x+u_{xxx}=0\). In case of a domain which is not invariant by translation with respect to \(x\) the relevant equation reads as \(u_t+u_x+uu_x+u_{xxx}=0\). It is proposed a well-posed mixed initial-boundary value problem in spatial domain of finite extent, \(u(0,t)=g(t)\), \(u_x(L,t)=h(t)\), \(u_{xx}(L,t)=k(t)\) for \(t\in [0,T)\) and \(u(x,0)=u_0(x)\) for \(x\in [0,L]\) (\(L>0, T\in (0,+\infty ]\)). The local existence of the solutions for arbitrary initial data in the Sobolev space \(H^1\) and global existence for small initial data are established. Global strong regularizing effects are investigated provided \(g=h=k=0\) in the sense that there exists a unique maximal weak solution which belongs to the class \(C([0,T \max);L^2)\cap L_{\text{loc}}^2([0,T \max);H^1)\). Some open questions are discussed.

35Q53 KdV equations (Korteweg-de Vries equations)
76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
35A07 Local existence and uniqueness theorems (PDE) (MSC2000)