Global Cauchy problem for the NLKG in super-critical spaces

By introducing a class of new function spaces $B_{p,q}^{\sigma ,s}$ as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon equation (NLKG) in all spatial dimensions $d⩾1$, $${\mathrm{\partial }}_t^{2}u+u-\mathrm{\Delta }u+u^{1+\alpha }=0,(u,{\mathrm{\partial }}_{t}u){\mathrm{\mid }}_{t=0}=(u_0,u_1)\mathrm{.}$$ We consider the initial data $(u_0,u_1)$ in super-critical function spaces $E^{\sigma ,s}\times E^{\sigma -1,s}$ for which their norms are defined by $$\mathrm{\parallel }f{\mathrm{\parallel }}_{E^{\sigma ,s}}=\mathrm{\parallel }⟨\xi {⟩}^{\sigma }{2}^{s\mathrm{\mid }\xi \mathrm{\mid }}\widehat{f}(\xi ){\mathrm{\parallel }}_{L^2},s<0,\sigma \in \mathbb{R}\mathrm{.}$$ Any Sobolev space $H^{\kappa }$ can be embedded into $E^{\sigma ,s}$, i.e., $H^{\kappa }\subset E^{\sigma ,s}$ for any $\kappa ,\sigma \in \mathbb{R}$ and $s<0$. We show the global existence and uniqueness of the solutions of NLKG if the initial data belong to some $E^{\sigma ,s}\times E^{\sigma -1,s}$ ($s<0,\sigma ⩾\max (d\mathrm{/}2-2\mathrm{/}\alpha ,1\mathrm{/}2),\alpha \in \mathbb{N},\alpha ⩾4\mathrm{/}d$) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in $E^{\sigma ,s}\times E^{\sigma -1,s}$ are not required for the global solutions. Similar results hold for the sinh-Gordon equation ${\mathrm{\partial }}_t^{2}u-\mathrm{\Delta }u+\sinh u=0$ if the spatial dimensions $d⩾2$.