Small solutions to nonlinear wave equations in the Sobolev spaces

The local and global well-posedness for the Cauchy problem for a class of nonlinear wave equations is studied. The global well-posedness of the problem is proved in the homogeneous Sobolev space Ḣ^s = Ḣ^s(ℝⁿ) of fractional order s > n/2 under the following assumptions: (1) Concerning the Cauchy data (φ,ψ) ∈ Ḣ ≡ Ḣ^s ⊕ Ḣ^s-1, ∥(φ,ψ); Ḣ^1/2∥ is relatively small with respect to ∥(φ,ψ); Ḣ^σ∥ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f, f(u) behaves as a power u^1+4/(n-1) near zero and has an arbitrary growth rate at infinity.