On The large Time Asymptotics of Klein-Gordon type equations with General Data-I

Abstract

We study the Klein-Gordon equation with general interaction term, which may be linear or nonlinear, and space-time dependent. The initial data is general, large and nonradial. We prove that global solutions are asymptotically given by a free wave and a weakly localized part. The proof is based on constructing in a new way the Free Channel Wave Operator, and further tools from the recent works [30,31,46,47]. This work generalizes the results of the first part of [30,31] on the Schrödinger equation to arbitrary dimension, and non-radial data.

On the Existence of Self-Similar solutions for some Nonlinear Schrödinger equations

“…) for all α ∈ (0, 1/3), where F c denotes a smooth characteristic function. When it comes to the case when n > 3, even though V(x, t) L ∞ t L 2 x (R n × R), by using a similar propagation estimate introduced in [27,28], we can get the existence of the free channel wave operator defined by (1.5), see Section 3. In [27,28], the weakly localized part ψ w,l (t) is defined by…”