Scattering for the cubic Klein–Gordon equation in two space dimensions

Abstract

Abstract. We consider both the defocusing and focusing cubic nonlinear Klein-Gordon equations utt − ∆u + u ± u 3 = 0 in two space dimensions for real-valued initial data u(0) ∈ H 1 x and ut(0) ∈ L 2 x . We show that in the defocusing case, solutions are global and have finite global L 4 t,x spacetime bounds. In the focusing case, we characterize the dichotomy between this behaviour and blowup for initial data with energy less than that of the ground state.These results rely on analogous statements for the two-dimensional cubic nonlinear Schrödinger equation, which are known in the defocusing case and for spherically-symmetric initial data in the focusing case. Thus, our results are mostly unconditional.It was previously shown by Nakanishi that spacetime bounds for KleinGordon equations imply the same for nonlinear Schrödinger equations.

Extremizers for Fourier restriction on hyperboloids

“…Note that supp(f 1 ) ⊆ {ξ ∈ R d : |ξ| ≤ 2} and supp(f N ) ⊆ {ξ ∈ R d : N 2 ≤ |ξ| ≤ 2N }, for N > 1. The following annular decoupling is in the spirit of [7,10].…”

“…Several authors have investigated the interface between bilinear restriction theory and these extremal questions, both from the restriction side and the partial differential equations point of view. Here we mention the works [1,2,5,6,7,10,14], all of which deal with these connections. Many other authors have contributed to the development of the area, and we refer the reader to [3] for an exposition of related literature on sharp Fourier restriction theory.…”

Scattering theory for the cubic nonlinear Klein–Gordon system

“…When the dimension n = 2, the system is mass critical. The scattering theory in this case remains open even if it is with the energy initial data; we refer the reader for the mass critical NLKG to Killip–Stovall–Visan . When the dimension n = 4, the system is energy critical, and we hope to return to the problem of proving the energy scattering theory for this system in a future work.…”

“…e L .u; t/ :D jP uj2 C jruj 2 C juj 2 , E L .u, v; t/ :D E L .u; t/ C E L .v; t/ D Z R L .u; t/ C e L .v; t/dx, G.u, v/ :D juj 2 jvj 2 , e N .u, v; t/ :D e L .u; t/ C e L .v; t/ C G.u, v/, E.u, v; t/ :D Z R L .u; t/ C 1 2 G.u, v/dx.Then the energy can be defined byE N .u, v; t/ :D Z R N .u, v; t/dx.…”

Untitled

“…See, for example, [Bahouri and Gérard 1999;Bahouri and Shatah 1998;Burq et al 2008;Burq and Planchon 2009;Ibrahim and Majdoub 2003;Ibrahim et al 2009;Kapitanski 1994; ; Killip et al 2012;Laurent 2011;Shatah and Struwe 1993;Tao 2006] for further discussion and references. In the case of the wave equation, passing to the variable coefficient setting is somewhat easier due the finite speed of propagation of solutions.…”