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

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 $s\ge n/2$. 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 $\exp (\kappa |u{|}^{\nu })$ with $\kappa >0$ and $0<\nu \le 2$ if $s=n/2$, and has an arbitrary growth rate if $s>n/2$. (2) Concerning the Cauchy data $(\varphi ,\psi )\in {\mathcal{H}}^s\equiv H^s\oplus H^{s-1}$, $\Vert (\varphi ,\psi );{\mathcal{H}}^{1/2}\Vert$ is relatively small with respect to $(\varphi ,\psi );{\mathcal{H}}^{s^{\ast }}$, where $s^{\ast }$ is a number with $s^{\ast }=n/2$ if $s=n/2$, $n/2<s^{\ast }\le s$ if $s>n/2$, and the smallness of $\Vert (\varphi ,\psi );{\dot{\mathcal{H}}}^{n/2}\Vert$ is also needed when $s=n/2$ and $\nu =2$.