Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis On unique continuation for nonlinear Schrödinger equations. (English) Zbl 1041.35072 Commun. Pure Appl. Math. 56, No. 9, 1247-1262 (2003). In this paper uniqueness properties of solutions to nonlinear Schrödinger equations of the form \[ \partial_t u+\Delta u+ F(u, u^*)= 0,\tag{1} \] with \((x, t)\in \mathbb{R}^n\times \mathbb{R}\), are investigated. Precisely, the authors consider the following problem: if \(u_1\) and \(u_2\) are solutions of (1) with \((x, t)\in \mathbb{R}^n\times [0,1]\), belonging to an appropriate class \(X\) and such that for some domain \(D\subset\mathbb{R}^n\), \(D\neq\mathbb{R}^n\), with \(u_1(x,0)= u_2(x,0)\), and \(u_1(x,1)= u_2(x,1)\), \(\forall x\in D\), is then \(u_1\equiv u_2\)? The authors answer the question in the of affirmative under very general assumptions on the nonlinearity expressed by the function \(F(u, u^*)\). The main result is represented by Theorem 1.1, whose proof is divided into a few steps. In this regards, in Lemma 2.1 an exponential decay estimate is proved. Then, Corollaries 2.2, 2.3, 2.4, Theorem 2.5 and Corollaries 2.6 and 2.7 are important aspects of the proof of Theorem 1.1. The paper is technically sound and could be appreciated by people engaged in researches addressed to nonlinear analysis and mathematical physics. Reviewer: Giulio Soliani (Lecce) Cited in 2 ReviewsCited in 40 Documents MSC: 35Q55 NLS equations (nonlinear Schrödinger equations) 35B60 Continuation and prolongation of solutions to PDEs Keywords:uniqueness; nonlinear Schrödinger equations; exponential decay estimate PDFBibTeX XMLCite \textit{C. E. Kenig} et al., Commun. Pure Appl. Math. 56, No. 9, 1247--1262 (2003; Zbl 1041.35072) Full Text: DOI References: [1] Bourgain, On the compactness of the support of solutions of dispersive equations, Internat Math Res Notices (9) pp 437– (1997) · Zbl 0882.35106 [2] Calderón, Commutators of singular integral operators, Proc Nat Acad Sci USA 53 pp 1092– (1965) · Zbl 0151.16901 [3] Isakov, Carleman type estimates in an anisotropic case and applications, J Differential Equations 105 (2) pp 217– (1993) · Zbl 0851.35028 [4] Hayashi, International Society for Analysis, Applications and Computation 9, in: Analytic extension formulas and their applications pp 59– (2001) [5] Kato, Advances in Mathematics Supplementary Studies 8, in: Studies in Applied Mathematics pp 93– (1983) [6] Kenig, On the support of solutions to the generalized KdV equation, Ann Inst H Poincaré Anal Non Linéaire 19 pp 191– (2002) · Zbl 1001.35106 [7] Kenig, Uniform Sobolev inequalities and unique continuation for second order constant coefficient differential operators, Duke Math J 55 (2) pp 329– (1987) · Zbl 0644.35012 [8] Kenig, A note on unique continuation for Schrödinger’s operator, Proc Amer Math Soc 103 (2) pp 543– (1988) · Zbl 0661.35056 [9] Saut, Unique continuation for some evolution equations, J Differential Equations 66 (1) pp 118– (1987) · Zbl 0631.35044 [10] Stein , E. M. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals Princeton Mathematical Series 43 Princeton University Press Princeton, N.J. 1993 [11] Tarama , S. Analytic solutions of the Korteweg-de Vries equation · Zbl 1078.35106 [12] Zhang, Unique continuation for the Korteweg-de Vries equation, SIAM J Math Anal 23 (1) pp 55– (1992) · Zbl 0746.35045 [13] Zhang, Unique continuation properties of the nonlinear Schrödinger equation, Proc Roy Soc Edinburgh Sect A 127 (1) pp 191– (1997) · Zbl 0879.35143 [14] Zuily, Uniqueness and non-uniqueness in the Cauchy problem (1983) · Zbl 0521.35003 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.