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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.

##### MSC:
 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)