On the global strong solutions of coupled Klein-Gordon-Schrödinger equations

The Cauchy problem for the coupled Klein–Gordon–Schrödinger system

“…Here u is a complex scalar nucleon field and v is a real scalar meson field. A large amount of work has been devoted to the study of Klein-GordonSchrödinger system [2,8,9,12,14,17,18,22], starting from I. Fukuda and M. Tsutsumi [9]. They considered the initial boundary value problem for the K-G-S system under the initial conditions u(0) = ϕ ∈ H (Ω) and the boundary conditions u(x, t) = v(x, t) = 0 for x ∈ ∂Ω and t ∈ R. Here Ω is a bounded smooth domain in R 3 .…”

Global Energy Solution to the Schrödinger Equation Coupled with the Chern-Simons Gauge and Neutral Field

“…We note that if we remove the gauge fields and the term | | 2 from the CSSn system, it is the same as the KleinGordon-Schödinger system with Yukawa coupling (KGS). There are many studies on the Cauchy problem of the KGS system in the Sobolev spaces [6][7][8][9]. Moreover, if we ignore the interaction with the neutral field which does not cause any difficulty in obtaining a local solution, a local solution for the CSSn system can be obtained in a similar way to the CSS system.…”

Wave Operators for the Coupled Klein-Gordon-Schrödinger Equations in Two Space Dimensions

“…We turn to the coupled Klein-Gordon-Schrö dinger equation. The time global well-posedness for the equation (KGS) is well-known (see [1,2,7,15]). Fukuda and M. Tsutsumi [8] and Strauss [36] studied the asymptotic behavior of the solutions to the coupled Klein-Gordon-Schrö dinger equations with interactions higher than the quadratic order.…”

“…CðR d þ dÞ a It is well-known that the equation (KGS) is globally well-posed in CðR; H 2 Þ l ½CðR; H 2 Þ V C 1 ðR; H 1 Þ (see, e.g., Baillon and Chadam [2], Fukuda and Tsutsumi [7] and Hayashi and von Wahl [15]). This implies that the unique local solution ðu; vÞ on ½T d ; yÞ, which is obtained above, can be extended to all times.…”