SMALL SOLUTIONS TO NONLINEAR WAVE EQUATIONS IN THE SOBOLEV SPACES

Abstract. 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 Ḣs = Ḣs(Rn) of fractional order s> n/2 under the following assumptions: (1) Concerning the Cauchy data (φ, ψ) ∈ Ḣs ≡ Ḣs ⊕ Ḣs−1, ‖(φ, ψ); Ḣ1/2 ‖ is relatively small with respect to ‖(φ, ψ); Ḣσ ‖ for any fixed σ with n/2 < σ ≤ s. (2) Concerning the nonlinearity f, f(u) behaves as a power u1+4/(n−1) near zero and has an arbitrary growth rate at infinity. 1