Normal Form and Global Solutions for the Klein–Gordon–Zakharov Equations

In the present paper we consider the Cauchy problem of the Klein-Gordon-Zakharov equations in three space dimensions: (1.1) ∂ t 2 u − Δ u + u = − n u , t > 0 , x ∈ R 3 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072096/6e6221c8-364d-4a5e-bfe5-98bb034e69b7/content/eq2539.tif"/> (1.2) ∂ t 2 n − Δ n = Δ | u | 2 , t > 0 , x ∈ R 3 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072096/6e6221c8-364d-4a5e-bfe5-98bb034e69b7/content/eq2540.tif"/> (1.3) u ( 0 , x ) = u 0 ( x ) , ∂ t u ( 0 , x ) = u 1 ( x ) , n ( 0 , x ) = n 0 ( x ) , ∂ t n ( 0 , x ) = n 1 ( x ) , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003072096/6e6221c8-364d-4a5e-bfe5-98bb034e69b7/content/eq2541.tif"/>