Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions

Abstract

In this paper, we consider the final state problem for the nonlinear Klein-Gordon equation (NLKG) with a critical nonlinearity in three space dimensions:We prove that for a given asymptotic profile uap, there exists a solution u to (NLKG) which converges to uap as t → ∞. Here the asymptotic profile uap is given by the leading term of the solution to the linear Klein-Gordon equation with a logarithmic phase correction. Construction of a suitable approximate solution is based on the combination of Fourier series expansion for the nonlinearity used in our previous paper [23] and smooth modification of phase correction by Ginibre and Ozawa [6].

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.…”

On asymptotic behavior of solutions to cubic nonlinear Klein-Gordon systems in one space dimension

“…A difference between this behavior and that for (1.3) is that the phase correction term Ψ ± depends on both Φ + and Φ − , which reflects the presence of an interaction between two components of the system. See also [15][16][17] for the two-and three-dimensional gauge invariant critical equation of the form (1.6).…”