Global Cauchy problem for the NLKG in super-critical spaces

Abstract

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 $d⩾1$, $${\partial }_t^{2}u+u-\Delta u+u^{1+\alpha }=0,(u,{\partial }_{t}u){|}_{t=0}=(u_0,u_1).$$ 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 $$\Vert f{\Vert }_{E^{\sigma ,s}}=\Vert ⟨\xi {⟩}^{\sigma }{2}^{s|\xi |}\hat{f}(\xi ){\Vert }_{L^2},s<0,\sigma \in \mathbb{R}.$$ 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 $E^{\sigma ,s}\times E^{\sigma -1,s}$ ($s<0,\sigma ⩾\max (d/2-2/\alpha ,1/2),\alpha \in \mathbb{N},\alpha ⩾4/d$) 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 ${\partial }_t^{2}u-\Delta u+\sinh u=0$ if the spatial dimensions $d⩾2$.

57 pages