JournalsprimsVol. 37, No. 3pp. 255–293

The Cauchy Problem for Nonlinear Klein–Gordon Equations in the Sobolev Spaces Klein-Gordon equations is studied in the Sobolev space Hs = Hs(

  • Makoto Nakamura

    Tohoku University, Sendai, Japan
  • Tohru Ozawa

    Hokkaido University, Sapporo, Japan
The Cauchy Problem for Nonlinear Klein–Gordon Equations in the Sobolev Spaces
Klein-Gordon equations is studied in the Sobolev space Hs = Hs( cover
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Abstract

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(ℝ_n_) with sn/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 _u_1+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) ∈ HsHsH__s-1, ‖(φ,ψ);H1/2‖ 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);Hn/2‖ is also needed when s = n/2 and ν = 2.

Cite this article

Makoto Nakamura, Tohru Ozawa, The Cauchy Problem for Nonlinear Klein–Gordon Equations in the Sobolev Spaces Klein-Gordon equations is studied in the Sobolev space Hs = Hs(. Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, pp. 255–293

DOI 10.2977/PRIMS/1145477225