The critical power of short pulse initial data on the global existence or blowup of smooth solutions to 3-D semilinear Klein-Gordon equations

It is well-known that there are global small data smooth solutions for the 3-D semilinear Klein-Gordon equations $\mathrm{\square }u+u=F(u,\mathrm{\partial }u)$ with cubic nonlinearities. However, for the short pulse initial data $(u,{\mathrm{\partial }}_{t}u)(0,x)=({\delta }^{\nu +1}{u}_0(\frac{x}{\delta }),{\delta }^{\nu }{u}_1(\frac{x}{\delta }))$ with $\nu \in \mathbb{R}$ and $(u_0,u_1)\in C_0^{\mathrm{\infty }}(\mathbb{R})$, which are a class of large initial data, we establish that when $\nu \le -\frac{1}{2}$, the solution $u$ can blow up in finite time for some suitable choices of $(u_0,u_1)$ and cubic nonlinearity $F(u,\mathrm{\partial }u)$; when $\nu >-\frac{1}{2}$, the smooth solution $u$ exists globally. Therefore, $\nu =-\frac{1}{2}$ is just the critical power corresponding to the global existence or blowup of smooth short pulse solutions for the cubic semilinear Klein-Gordon equations.