一类具阻尼项的Klein Gordon方程解的衰减行为

On the Asymptotic Behavior of Solution for the Klein-Gordon Equation with Damped Term

讨论了一类带阻尼项的n维Klein Gordon方程的柯西问题,观察到其线性方程的耗散结构是正则耗散型,这将意味着在对初值的正则假设下,可以得到解的最佳衰减估计.基于其相应线性方程的衰减估计和小初值条件,利用压缩映射原理,在Sobolev空间中证明了整体解的存在性和小振幅解的渐近行为.

In this paper, we study the Cauchy problem for the Klein-Gordon equation with damped term in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. Based on the decay estimates of solutions to the corresponding linear equation and smallness condition on the initial data,we prove the global existence and asymptotic of the small amplitude solution in the time-weighted Sobolev space by the contraction mapping principle.