Critical nonlinear Schrödinger equations in higher space dimensions. (English) Zbl 1408.35173

Hayashi, Nakao; Li, Chunhua; Naumkin, Pavel I.

J. Math. Soc. Japan 70, No. 4, 1475-1492 (2018).

Summary: We study the critical nonlinear Schrödinger equations \[ i\partial _{t}u+\frac{1}{2}\Delta u = \lambda | u|^{{2}/{n}}u, (t,x) \in \mathbb{R}^{+}\times \mathbb{R}^{n}, \] in space dimensions \(n\geq 4\), where \(\lambda \in \mathbb{R}\). We prove the global in time existence of solutions to the Cauchy problem under the assumption that the absolute value of Fourier transform of the initial data is bounded below by a positive constant. Also we prove the two side sharp time decay estimates of solutions in the uniform norm.

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
35B40	Asymptotic behavior of solutions to PDEs
35A01	Existence problems for PDEs: global existence, local existence, non-existence

Keywords:

critical NLS equations; higher space dimensions; large time asymptotics

References:

[1]	R. Carles, Geometric optics and long range scattering for one dimensional nonlinear Schrödinger equations, Commun. Math. Phys., 220 (2001), 41–67. · Zbl 1029.35211
[2]	Y. Cho and T. Ozawa, On the semirelativistic Hartree-type equation, SIAM J. Math. Anal., 38 (2006), 1060–1074. · Zbl 1122.35119
[3]	J.-M. Delort, Existence globale et comportement asymptotique pour l’équation de Klein–Gordon quasi-linéaire à données petites en dimension 1, Ann. Sci. École Norm. Sup. (4), 34 (2001), 1–61.
[4]	J. Ginibre and T. Ozawa, Long range scattering for nonlinear Schrödinger and Hartree equations in space dimension \(n≥ 2\), Commun. Math. Phys., 151 (1993), 619–645. · Zbl 0776.35070
[5]	N. Hayashi, C. Li and P. I. Naumkin, Nonlinear Schrödinger systems in 2d with nondecaying final data, J. Differential Equations, 260 (2016), 1472–1495. · Zbl 1328.35211
[6]	N. Hayashi and P. I. Naumkin, Asymptotics in large time of solutions to nonlinear Schrödinger and Hartree equations, Amer. J. Math., 120 (1998), 369–389. · Zbl 0917.35128
[7]	N. Hayashi and P. I. Naumkin, Domain and range of the modified wave operator for Schrödinger equations with a critical nonlinearity, Commun. Math. Phys., 267 (2006), 477–492. · Zbl 1113.81121
[8]	N. Hayashi and P. I. Naumkin, The initial value problem for the cubic nonlinear Klein–Gordon equation, Zeitschrift fur Angewandte Mathematik und Physik, 59 (2008), 1002–1028. · Zbl 1190.35199
[9]	N. Hayashi and P. I. Naumkin, Final state problem for the cubic nonlinear Klein–Gordon equation, J. Math. Phys., 50 (2009), 103511, 14pp. · Zbl 1283.35056
[10]	T. Kato, On nonlinear Schrödinger equations II, \(H^s\)-solutions and unconditional wellposedness, J. Anal. Math., 67 (1995), 281–306.
[11]	C. Li and N. Hayashi, Critical nonlinear Schrödinger equations with data in homogeneous weighted \(\mathbf{L}^{2}\) spaces, J. Math. Anal. Appl., 419 (2014), 1214–1234. · Zbl 1296.35171
[12]	T. Ozawa, Long range scattering for nonlinear Schrödinger equations in one space dimension, Commun. Math. Phys., 139 (1991), 479–493. · Zbl 0742.35043