Exponential Energy Decay for Damped Klein–Gordon Equation with Nonlinearities of Arbitrary Growth
Exponential Energy Decay for Damped Klein–Gordon Equation with Nonlinearities of Arbitrary Growth
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DOI：
10.1080/03605302.2010.534684
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发表时间：
2010-01
影响因子：
1.9
通讯作者：
Lassaad Aloui;S. Ibrahim;Kenji Nakanishi
中科院分区：
文献类型：
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作者：
Lassaad Aloui;S. Ibrahim;Kenji Nakanishi
我们得到了一个均匀的指数衰减的非线性Klein-Gordon方程的总能量的阻尼在空间无穷大附近的cBN或在外部的一个星形障碍。Zuazua [37，38]首先证明了这种结果，用于适度增长的散焦非线性，后来Dehman等人[7]使用线性近似和唯一的连续参数扩展到能量亚临界情况。我们提出了一种不同的方法，完全基于Morawetz型先验估计，它适用于散焦非线性的任意增长，包括能量临界的情况下，超临界的情况下，在任何尺寸的指数非线性。我们的证明的一个优点，即使在温和的增长的情况下，是衰减率是独立的非线性。一旦我们控制了Morawetz型估计中的非线性部分，我们也可以处理那些能量小于基态能量的解的聚焦情况。特别地，当我们有无阻尼方程的散射时，这可以实现。
We derive a uniform exponential decay of the total energy for the nonlinear Klein–Gordon equation with a damping around spatial infinity in ℝ N or in the exterior of a star-shaped obstacle. Such a result was first proved by Zuazua [37, 38] for defocusing nonlinearity with moderate growth, and later extended to the energy subcritical case by Dehman et al. [7], using linear approximation and unique continuation arguments. We propose a different approach based solely on Morawetz-type a priori estimates, which applies to defocusing nonlinearity of arbitrary growth, including the energy critical case, the supercritical case and exponential nonlinearities in any dimensions. One advantage of our proof, even in the case of moderate growth, is that the decay rate is independent of the nonlinearity. We can also treat the focusing case for those solutions with energy less than the one of the ground state, once we get control of the nonlinear part in Morawetz-type estimates. In particular this can be achieved when we have the scattering for the undamped equation.