(t-t^{'}) \langle D \rangle \right)}{\langle D \rangle} |u|^{p-1}(t^{'}) u(t^{'}) \, dt^{'}
\end{array}
\label{Eqn:StrongSol}$$

The $p$- defocusing Klein-Gordon equation is closely related to the $p$- defocusing wave equation i.e

$$\begin{array}{ll}
\partial_{tt} v - \triangle v = & - |v|^{p-1} v
\end{array}
\label{Eqn:DefocPWaveEq}$$

with data $v(0)=v_{0}$, $\partial_{t} v(0)=v_{1}$. (\[Eqn:DefocPWaveEq\]) enjoys the following scaling property

$$\begin{array}{ll}
v(t,x) & \rightarrow \frac{1}{\lambda^{\frac{2}{p-1}}} u \left( \frac{t}{\lambda}, \frac{x}{\lambda} \right) \\
v_{0}(x) & \rightarrow \frac{1}{\lambda^{\frac{2}{p-1}}} u_{0}  \left( \frac{x}{\lambda} \right) \\
v_{1}(x) & \rightarrow \frac{1}{\lambda^{\frac{2}{p-1}+1}} u_{1} \left( \frac{x}{\lambda} \right)
\end{array}
\label{Eqn:WaveScaling}$$

We define the critical exponent $s_{c}:= \frac{3}{2} - \frac{2}{p-1}$. One can check that the $\dot{H}^{s_{c}} \times \dot{H}^{s_{c}-1}$ norm of $(u_{0},u_{1})$ is invariant under the transformation (\[Eqn:WaveScaling\]) [^1]. (\[Eqn:DefocPWaveEq\]) was demonstrated to be locally well-posed in [@linsog] and in $H^{s} \times H^{s-1}$, $s >\frac{3}{2} - \frac{2}{p-1}$, $p>3$ by using an iterative argument. In fact their results extend immediately to (\[Eqn:NlkgWdat\]) [^2].

If $p=5$ then $s_{c}=1$ and this is why we say that that the nonlinearity $|u|^{p-1} u$ is $\dot{H}^{1}$ critical. If $3<p<5$ then $s_{c}<1$ and the regime is $\dot{H}^{1}$ subcritical.

It is well-known that smooth solutions to (\[Eqn:NlkgWdat\]) have a conserved energy

$$\begin{array}{ll}
E(u(t)) &  := \frac{1}{2} \int_{\mathbb{R}^{3}} \left| \partial_{t} u (t,x) \right|^{2} \, dx  + \frac{1}{2} \int_{\mathbb{R}^{3}} | 