Birnir, Björn; Kenig, Carlos E.; Ponce, Gustavo; Svanstedt, Nils; Vega, Luis On the ill-posedness of the IVP for the generalized Korteweg-de Vries and nonlinear Schrödinger equations. (English) Zbl 0855.35112 J. Lond. Math. Soc., II. Ser. 53, No. 3, 551-559 (1996). 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. Reviewer: N.Svanstedt (Santa Barbara) Cited in 41 Documents 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 Keywords:generalized Korteweg-de Vries equation; nonlinear Schrödinger equation; ill-posedness; continuous dependence; well-posedness PDFBibTeX XMLCite \textit{B. Birnir} et al., J. Lond. Math. Soc., II. Ser. 53, No. 3, 551--559 (1996; Zbl 0855.35112) Full Text: DOI