Global Cauchy problem for the complex NLKG and sinh-Gordon equations in super-critical spaces

By introducing a class of new function spaces Bp,qσ,s as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon equation (NLKG) and sinh-Gordon in all spatial dimensions d⩾1,∂t2u+u−Δu+f(u)=0,(u,∂tu)|t=0=(u0,u1), where f(u)=u1+α or f(u)=sinh⁡u−u. We consider the initial data (u0,u1) in super-critical function spaces Eσ,s×Eσ−1,s for which their norms are defined by‖f‖Eσ,s=‖〈ξ〉σ2s|ξ|fˆ(ξ)‖L2,s<0,σ∈R. Any Sobolev space Hκ can be embedded into Eσ,s, i.e., Hκ⊂Eσ,s for any κ,σ∈R and s<0. We show the global existence and uniqueness of the solutions of NLKG if the initial data belong to some Eσ,s×Eσ−1,s (s<0,σ⩾max⁡(d/2−2/α,1/2),α∈N,α⩾4/d) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in Eσ,s×Eσ−1,s are not required for the global solutions. Similar results hold for the sinh-Gordon equation if the spatial dimensions d⩾2.