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Instabilities for supercritical Schrödinger equations in analytic manifolds. (English) Zbl 1157.35107
This paper concerns with the problem of well-posedness of nonlinear evolutionary PDE’s. Usually people working in functional analysis, fix some functional space where to choice initial data and to find solutions, then look if for such initial conditions there corresponds a unique solution uniformly continuous in the time variable. Typically Sobolev spaces \(H^s\) are considered. Then the study is concentred on a critical value \(s_{cr}\) such that for \(s>s_{cr}\) one has well-posedness. Here the author is interested in the following Cauchy problem for a Schrödinger type equation:
\[ \begin{cases} i\partial_tu+\triangle_gu=\omega| u| ^{p-1}u,\quad (t,x)\in{\mathbb R}\times M,\\ u(0,x)=u_0(x),\end{cases} \] where \(M\) is a \((d\geq 3)\)-dimensional analytic Riemannian manifold, \(\omega\in\{-1,1\}\), \(p\) is an odd integer. (The classification used here of “sub-critical case”, “’critical case” and “’super-critical case”, refers to some particular ranges for parameters. See literature quoted in the paper. [1–5].) The main result is that in the supercritical case the nonlinear self-interaction of the wave leads to some instability of the waves (solutions).
Reviewer’s remark: The approach considered in this paper to solve such Cauchy problems follows the usual functional analysis one. Unfortunately in this way one forgets solutions with non trivial topology that, instead, are the more interesting ones to consider especially by considering the physical phenomena that such equations should describe.

MSC:
35Q55 NLS equations (nonlinear Schrödinger equations)
58J05 Elliptic equations on manifolds, general theory
35B33 Critical exponents in context of PDEs
35B35 Stability in context of PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
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References:
[1] T. Alazard, R. Carles, Sequential loss of regularity for super-critical nonlinear Schrödinger equations, preprint
[2] Baouendi, M.S.; Goulaouic, C., Remarks on the abstract form of nonlinear cauchy – kovalevsky theorems, Comm. partial differential equations, 2, 11, 1151-1162, (1977) · Zbl 0391.35006
[3] N. Burq, P. Gérard, S. Ibrahim, Ill-posedness for supercritical non-linear Schrödinger and wave equations, preprint
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[10] Grenier, E., Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. amer. math. soc., 126, 2, 523-530, (1998) · Zbl 0910.35115
[11] Lebeau, G., Nonlinear optic and supercritical wave equation, Bull. soc. roy. sci. liège, 70, 4-6, 267-306, (2001) · Zbl 1034.35137
[12] Lebeau, G., Perte de régularité pour LES équations d’ondes sur-critiques, Bull. soc. math. France, 133, 1, 145-157, (2005) · Zbl 1071.35020
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