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) ∈ Ḣ ≡ 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.
|Number of pages||20|
|Journal||Houston Journal of Mathematics|
|Publication status||Published - 2001 Dec 1|
- Besov spaces
- Sobolev spaces
- Wave equations
ASJC Scopus subject areas