SIAM Journal on Mathematical Analysis, Volume 52, Issue 6, Page 6379-6411, January 2020.
We obtain sharp decay estimates and asymptotics for small solutions to the one-dimensional Klein--Gordon equation with constant coefficient cubic and spatially localized, variable coefficient cubic nonlinearities. Vector-field techniques to deal with the long-range nature of the cubic nonlinearity become problematic in the presence of variable coefficients. We introduce a novel approach based on pointwise-in-time local decay estimates for the Klein--Gordon propagator to overcome this impasse.

中文翻译：

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变系数三次非线性一维Klein-Gordon方程的衰减与渐近性

SIAM数学分析杂志，第52卷，第6期，第6379-6411页，2020年1月。
我们获得了一维Klein-Gordon方程的小解的锐变衰减估计和渐近性，该方程具有常数系数立方和空间局部变量系数三次非线性。在存在可变系数的情况下，用于处理立方非线性的长距离性质的矢量场技术变得成问题。我们针对Klein-Gordon传播器引入了基于时间点局部衰减估计的新颖方法来克服这种僵局。