Global dynamics below the ground state for the focusing semilinear Schrodinger equation with a linear potential

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:503

Article

Ikeda, Masahiro^1,2 

[1] Keio Univ, Fac Sci & Technol, Dept Math, Kohoku Ku, 3-14-1 Hiyoshi, Yokohama, Kanagawa 2238522, Japan

[2] RIKEN, Ctr Adv Intelligence Project, Tokyo, Japan

关键词: Scattering;    Blow-up;    Ground state;    Semilinear Schrodinger equation;    Linear potential;   

DOI  :  10.1016/j.jmaa.2021.125291

来源: Elsevier

【 摘 要 】

We study global dynamics of the solution to the Cauchy problem for the focusing semilinear Schrodinger equation with a linear potential on the real line R: {i partial derivative(t)u + partial derivative(2)(x)u - Vu + vertical bar u vertical bar(p-1) u = 0, (t, x) is an element of I x R, u(0) = u(0) is an element of H, (NLSV) where u = u(t, x) is a complex-valued unknown function of (t, x) is an element of I x R, I denotes the maximal existence time interval of u, V = V(x) is non-negative and in L-1(R) + L-infinity(R), pbelongs to the so-called mass-supercritical case, i.e. p > 5, and His a Hilbert space connected to the Schrodinger operator -partial derivative(2)(x) + V and is called energy space. It is well known that (NLSV) is locally well-posed in H. Our aim in the present paper is to study global behavior of the solution and prove a scattering result and a blow-up result for (NLSV) with the data u(0) whose mass-energy is less than that of the ground state Q, where the function Q = Q(x) is the unique radial positive solution to the stationary Schrodinger equation without the potential: -Q '' + Q - vertical bar Q vertical bar(p-1)Q, in H-1(R). The similar result for NLS without potential (V equivalent to 0), which is invariant of translation and scaling transformation, in one space dimension was obtained by Akahori-Nawa. Lafontaine treated the defocusing version of (NLSV), that is, (NLSV) with a replacement of +vertical bar u vertical bar(p-1)u into -vertical bar u vertical bar(p-1)u, and prove that the solution scatters as t ->+/-infinity in H-1(R) for an arbitrary data in H-1(R) by Kenig-Merle's argument with a profile decomposition. However, the method to the defocusing case cannot be applicable to our focusing case because the energy is positive in the defocusing case, on the other hand, the energy may be negative in the focusing case. To overcome this difficulty, we use a variational argument. Our proof of the blow-up result is based on the argument of Du-Wu-Zhang. The difficulty of our case lies in deriving a uniform bound of a functional related to Virial Identity because of existence of the potential. (C) 2021 Elsevier Inc. All rights reserved.

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