The Cauchy Problem for Nonlinear Klein–Gordon Equations in the Sobolev Spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space H = H( ) with s ≥ n/2. The global well-posedness of the problem is proved under the following assumptions: (1) Concerning the nonlinearity f , f(u) behaves as a power u near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|ν ) with κ > 0 and 0 n/2. (2) Concerning the Cauchy data (φ,ψ) ∈ Hs ≡ H ⊕ Hs−1, ‖(φ, ψ);H1/2‖ is relatively small with respect to ‖(φ, ψ); Ḣs‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 n/2, and the smallness of ‖(φ, ψ); Ḣn/2‖ is also needed when s = n/2 and ν = 2. §