Global Cauchy problem for the NLKG in super-critical spaces

2023/03/10 by Wang, Baoxiang
#35L71 #42B35 #42B37 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA)

paper · doi:10.48550/arxiv.2303.05969

Abstract

By introducing a class of new function spaces B^σ,s_p,q as the resolution spaces, we study the Cauchy problem for the nonlinear Klein-Gordon equation (NLKG) in all spatial dimensions d \geqslant 1, ∂²_t u + u- Δu + u^1+α =0, (u, ∂_t u)|_t=0 = (u₀,u₁). We consider the initial data (u₀,u₁) in super-critical function spaces E^σ,s × E^σ-1,s for which their norms are defined by ‖f‖_E^σ,s = ‖⟨ξ⟩^σ2^s|ξ|\widehatf(ξ)‖_L², slt;0, σ∈ ℝ. Any Sobolev space H^κ can be embedded into E^σ,s, i.e., H^κ⊂ E^σ,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 E^σ,s × E ^σ-1,s (s<0, σ\geqslant max (d/2-2/α, 1/2), α∈ ℕ, α\geqslant 4/d) and their Fourier transforms are supported in the first octant, the smallness conditions on the initial data in E^σ,s × E^σ-1,s are not required for the global solutions. Similar results hold for the sinh-Gordon equation ∂²_t u - Δu + \sinh u=0 if the spatial dimensions d \geqslant 2.

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