# 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*);

**1/2‖ is relatively small with respect to (**

*H**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