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^s = H^s(Rⁿ) 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 u¹⁺⁴/ⁿ near zero. At infinity f(u) has an exponential growth rate such as exp(κ|u|^ν) with κ > 0 and 0<ν≤2 if s = n/2, and has an arbitrary growth rate if s > n/2. (2) Concerning the Cauchy data (ϕ,Ψ) e H^s = H^s ⊕ H^S-1, ‖(ϕ,Ψ); H^1/2‖ is relatively small with respect to ‖(ϕ,Ψ); H^s∗‖, where s∗ is a number with s∗ = n/2 if s = n/2, n/2 < s∗ ≤ s if s > n/2, and the smallness of ‖(ϕ,Ψ); H^n/2‖ is also needed when s = n/2 and ν = 2.