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 (NLKG) and sinh-Gordon equations 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 da...

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Bibliographic Details
Published in:Journal of functional analysis 2024-07, Vol.287 (2), p.110458, Article 110458
Main Author: Wang, Baoxiang
Format: Article
Language:English
Subjects:
Global solution
NLKG
Sinh-Gordon
Very rough function spaces
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Summary: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 (NLKG) and sinh-Gordon equations 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
ISSN:0022-1236
1096-0783
DOI:10.1016/j.jfa.2024.110458