# 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

## 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 *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 _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*) ∈ * Hs* ≡

*Hs*⊕

*H__s*-1, ‖(

*φ,ψ*);

*H***1/2‖ is relatively small with respect to (***φ, ψ*);**_s_∗, where _s_∗ is a number with _s_∗ =***H**n*/2 if*s*=*n*/2,*n*/2 < _s_∗ ≤*s*if*s*>*n*/2, and the smallness of ‖(*φ, ψ*);*H**n*/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