Long range scattering for the complex-valued Klein-Gordon equation with quadratic nonlinearity in two dimensions

Abstract

In this paper, we study large time behavior of complexvalued solutions to nonlinear Klein-Gordon equation with a gauge invariant quadratic nonlinearity in two spatial dimensions. To find a possible asymptotic behavior, we consider the final value problem. It turns out that one possible behavior is a linear solution with a logarithmic phase correction as in the real-valued case. However, the shape of the logarithmic correction term has one more parameter which is also given by the final data. In the real case the parameter is constant so one cannot see its effect. However, in the complex case it varies in general. The one dimensional case is also discussed.

Scattering for the mass-critical nonlinear Klein-Gordon equations in three and higher dimensions

“…Remark 5.2. The integral (5.2) also appears in the work of the third author and his collaborators [39][40][41][42].…”

“…In this subsection, we give another proof of (5.11) in Theorem 3.2 motivated by the argument in [39]. We also refer to the recent works [40][41][42] on the quadratic NLKG equations, where a similar argument is used. A main ingredient is the Fourier series expansion By means of the Strichartz estimate, one has the desired estimate…”

“…In the higher dimensions, there is one difficulty that the power of the nonlinear term is fractional order, and we cannot write the limit system clearly as in one and two dimensional case. Fortunately, by the technique developed by the third author and his collaborators [37][38][39][40][41][42], we can give the limit system at least formally, which is still the mass-critical NLS. Thus we can still use the solution of the mass-critical NLS equation to approximate the large scale profile.…”

“…One way is inspired by the work of the nonrelativistic limit of the NLKG equation in [36,43,48], and we deal with the nonlinear term by using some generalized integration by parts formula. The other way is to follow the argument in [37][38][39][40][41][42], and especially [40,41]. In these works, they developed a very powerful tools to deal with the nonlinear dispersive equations with non-algebraic nonlinearity.…”

Optimal decay rate of solutions for nonlinear Klein-Gordon systems of critical type

“…At the same time, Sunagawa [17] shows that there exists a solution which decays like O(t − 1 2 ) in L ∞ , the same decay as a free solution, but does not behave like a free solution and exhibits a modified-scattering type behavior. Masaki-Segata-Uriya [11] show modified scattering in the complex-valued case for d = 1, 2. Remark that the single complex-valued equation is equivalent to a system of real-valued system (see also [19]).…”