Almost global smooth solutions of the 3D quasilinear Klein-Gordon equations on the product space R^(2)×T  

作　　者：Jun Li Fei Tao Huicheng Yin 

机构地区：[1]Department of Mathematics,Nanjing University,Nanjing 210093,China [2]School of Mathematical Sciences and Mathematical Institute,Nanjing Normal University,Nanjing 210023,China

出　　处：《Science China Mathematics》2024年第12期2713-2752,共40页中国科学（数学英文版）

基　　金：supported by National Natural Science Foundation of China(Grant No.11871030);supported by National Natural Science Foundation of China(Grant No.11731007)。

摘　　要：In this paper,for the 3D quadratic nonlinear Klein-Gordon equation on the product space R^(2)×T,we focus on the lower bound of the lifespan of the smooth solution with slowly decaying initial data.When the size of initial data is bounded by ε_(0)>0,it is shown that a smooth solution exists up to the time C_(0)/20 with0 being sufficiently small and e^(c0)/ε_(0)^(2)>0 being some suitable constant.Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on R^(2)×T only admits the optimal time-decay rate(1+t)−1,from which we generally derive the lifespan of the nonlinear Klein-Gordon equation up to e^(c0/ε0)rather than the more precise e^(c0/ε^(2)0) here.

关 键 词：quasilinear Klein-Gordon equation almost global solution Z-norm Littlewood-Paley decomposition space-time resonance method energy estimate 

分 类 号：O175.29[理学—数学]