Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities
Asymptotics for 1D Klein-Gordon Equations with Variable Coefficient Quadratic Nonlinearities
复制标题
DOI：
10.1007/s00205-021-01675-y
复制
发表时间：
2020-06
影响因子：
2.5
通讯作者：
Hans Lindblad;Jonas Lührmann;A. Soffer
中科院分区：
文献类型：
--
作者：
Hans Lindblad;Jonas Lührmann;A. Soffer
本文研究了具有变系数二次非线性项的一维Klein-Gordon方程小解的渐近性态。在这项工作中的主要发现是一个惊人的共振之间的特定空间频率的可变系数和时间振荡的解决方案。在共振的情况下，一种新型的修改散射行为发生，表现出对数减慢的衰减率沿沿着某些射线。在非共振的情况下，我们引入了一个新的变系数二次规范形式，并建立了尖锐的衰减估计和渐近的存在下，一个临界分散的常系数立方非线性。本文所考虑的Klein-Gordon模型是由研究经典非线性标量场方程在真实的直线上的扭解的渐近稳定性而产生的。
We initiate the study of the asymptotic behavior of small solutions to one-dimensional Klein-Gordon equations with variable coefficient quadratic nonlinearities. The main discovery in this work is a striking resonant interaction between specific spatial frequencies of the variable coefficient and the temporal oscillations of the solutions. In the resonant case a novel type of modified scattering behavior occurs that exhibits a logarithmic slow-down of the decay rate along certain rays. In the non-resonant case we introduce a new variable coefficient quadratic normal form and establish sharp decay estimates and asymptotics in the presence of a critically dispersing constant coefficient cubic nonlinearity. The Klein-Gordon models considered in this paper are motivated by the study of the asymptotic stability of kink solutions to classical nonlinear scalar field equations on the real line.