首页|Almost global smooth solutions of the 3D quasilinear Klein-Gordon equations on the product space R2 × T
Almost global smooth solutions of the 3D quasilinear Klein-Gordon equations on the product space R2 × T
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In this paper,for the 3D quadratic nonlinear Klein-Gordon equation on the product space R2 × 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 ∈o＞0,it is shown that a smooth solution exists up to the time ec0/∈20 with ∈0 being sufficiently small and c0＞0 being some suitable constant.Note that the solution of the corresponding 3D linear homogeneous Klein-Gordon equation on R2 × 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 ec0/∈0 rather than the more precise ec0/∈20 here.
quasilinear Klein-Gordon equationalmost global solutionZ-normLittlewood-Paley decomposi-tionspace-time resonance methodenergy estimate
Jun Li、Fei Tao、Huicheng Yin
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Department of Mathematics,Nanjing University,Nanjing 210093,China
School of Mathematical Sciences and Mathematical Institute,Nanjing Normal University,Nanjing 210023,China
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