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On unique continuation for nonlinear Schrödinger equations. (English) Zbl 1041.35072
In this paper uniqueness properties of solutions to nonlinear Schrödinger equations of the form $$\partial_t u+\Delta u+ F(u, u^*)= 0,\tag1$$ with $(x, t)\in \bbfR^n\times \bbfR$, are investigated. Precisely, the authors consider the following problem: if $u_1$ and $u_2$ are solutions of (1) with $(x, t)\in \bbfR^n\times [0,1]$, belonging to an appropriate class $X$ and such that for some domain $D\subset\bbfR^n$, $D\ne\bbfR^n$, 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.

35Q55NLS-like (nonlinear Schrödinger) equations
35B60Continuation of solutions of PDE
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