# The Cauchy Problem for Nonlinear Klein–Gordon Equations in the Sobolev Spaces

### 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 $H_{s}=H_{s}(R_{n})$ 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 $(ϕ,ψ)∈H_{s}≡H_{s}⊕H_{s−1}$, $‖(ϕ,ψ);H_{1/2}‖$ 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 $‖(ϕ,ψ);H˙_{n/2}‖$ 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. Publ. Res. Inst. Math. Sci. 37 (2001), no. 3, pp. 255–293

DOI 10.2977/PRIMS/1145477225