The local and global well-posedness for the Cauchy problem for a class of nonlinear Klein-Gordon equations is studied in the Sobolev space Hs = Hs(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 Hs = Hs ⊕ HS-1, ‖(ϕ,Ψ); H1/2‖ is relatively small with respect to ‖(ϕ,Ψ); Hs∗‖, 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); Hn/2‖ is also needed when s = n/2 and ν = 2.
|ジャーナル||Publications of the Research Institute for Mathematical Sciences|
|出版物ステータス||Published - 2001|
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