Long range scattering and modified wave operators for some Hartree type equations. III: Gevrey spaces and low dimensions (Q5949600)

The authors study the scattering theory for a class of Hartree type equations NEWLINE\[NEWLINEi\partial_t u+ {1\over 2}\Delta u = kt^{\mu-\gamma}|\nabla|^{\mu-n}|u|^2u NEWLINE\]NEWLINE in \({\mathbb R}^n, n \geq 1\) with \(0<\gamma\leq 1\) and \(0<\mu<n\). NEWLINENEWLINENEWLINEThe existence of modified local wave operators at infinity with no size restriction of the data is proved. The asymptotic behaviour in time of solutions in the range of the wave operators is investigated. NEWLINENEWLINENEWLINEThis paper is the third in the series of the papers [see Part I, Rev. Math. Phys. 12, No. 3, 361-429 (2000; Zbl 1044.35041); Part II, Ann. Henri Poincaré 1, No. 4, 753-800 (2000; Zbl 1024.35084)]. The paper is self-contained and extends results of the previous papers. The proofs are based on the framework of Gevrey spaces.