TY - JOUR
T1 - Remark on the global existence and large time asymptotics of solutions for the quadratic NLS
AU - Hayashi, Nakao
AU - Naumkin, Pavel I.
N1 - Funding Information:
The work of N.H. is partially supported by KAKENHI (no. 19340030 ) and the work of P.I.N. is partially supported by CONACYT and PAPIIT .
Copyright:
Copyright 2011 Elsevier B.V., All rights reserved.
PY - 2011/12
Y1 - 2011/12
N2 - We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions i ∂tu+12Δu=|u|2,t>0,x R4,u(0,x)= u0(x),x R4. We prove that if the initial data u0 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.
AB - We study the global in time existence of small solutions to the nonlinear Schrödinger equation with quadratic interactions i ∂tu+12Δu=|u|2,t>0,x R4,u(0,x)= u0(x),x R4. We prove that if the initial data u0 satisfy smallness conditions in the weighted Sobolev norm, then the solution of the Cauchy problem (0.1) exists globally in time. Furthermore, we prove the existence of the usual scattering states and find the large time asymptotics of the solutions.
KW - Global existence
KW - Nonlinear Schrödinger equations
KW - Quadratic nonlinearities
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U2 - 10.1016/j.na.2011.07.016
DO - 10.1016/j.na.2011.07.016
M3 - Article
AN - SCOPUS:80052797541
VL - 74
SP - 6950
EP - 6964
JO - Nonlinear Analysis, Theory, Methods and Applications
JF - Nonlinear Analysis, Theory, Methods and Applications
SN - 0362-546X
IS - 18
ER -