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On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations. (English) Zbl 0855.35112
The initial value problem (IVP) for the generalized Korteweg-de Vries (GKdV) equation \[ \partial_t u+ \partial^3_x u+ u^k \partial_x u= 0, \quad t, x\in \mathbb{R},\quad k\in \mathbb{Z}^+,\quad u(x, 0)= u_0(x) \] and the nonlinear Schrödinger (NLS) equation \[ i\partial_t u+ \Delta u+ \lambda|u|^\alpha u= 0,\quad t\in \mathbb{R},\quad x\in \mathbb{R}^n, \lambda> 0, \alpha> 0, u(x, 0)= u_0(x) \] are studied. For the GKdV equation the authors prove that for \(k\geq 4\) the IVP is ill-posed for data \(u_0\) in the Sobolev space \(H^{s_k}(\mathbb{R})= (1- \Delta)^{- s_k/2} L^2(\mathbb{R})\), where \(s_k= 1/2- 2/k\). For the NLS equation they prove that for \(4/n\leq \alpha< \infty\) the IVP is ill-posed in \(H^{s_\alpha}(\mathbb{R}^n)\), where \(s_\alpha= n/2- 2/\alpha\). In both cases the ill-posedness is proved in the sense that the time of existence and the continuous dependence cannot be expressed in terms of the size of the data in the norm of these spaces. These results in particular show that the well-posedness results in these critical cases are sharp.

MSC:
35Q53 KdV equations (Korteweg-de Vries equations)
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35R25 Ill-posed problems for PDEs
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