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 .

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

SN - 0362-546X

VL - 74

SP - 6950

EP - 6964

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

IS - 18

ER -