Zeitschrift für angewandte Mathematik und Physik ( IF 1.7 ) Pub Date : 2020-08-27 , DOI: 10.1007/s00033-020-01373-x Ruy Coimbra Charão , Ryo Ikehata
We introduce a new model of the nonlocal wave equation with a logarithmic damping mechanism, which is rather weak as compared with frequently studied fractional damping cases. We consider the Cauchy problem for the new model in \(\mathbf{R}^{n}\) and study the asymptotic profile and optimal decay rates of solutions as \(t \rightarrow \infty \) in \(L^{2}\)-sense. The damping terms considered in this paper is not studied so far, and in the low-frequency parameters, the damping is rather weakly effective than that of well-studied power type one such as \((-\Delta )^{\theta }u_{t}\) with \(\theta \in (0,1)\). When getting the optimal rate of decay, we meet the so-called hypergeometric functions with special parameters, so the analysis seems to be more difficult and attractive.
中文翻译:
具有对数阻尼的某些波动方程解的渐近分布和最优衰减
我们引入了具有对数阻尼机制的非局部波动方程的新模型,与经常研究的分数阻尼情况相比,该模型较弱。我们考虑\(\ mathbf {R} ^ {n} \)中新模型的柯西问题,并以\(t \ rightarrow \ infty \)在\(L ^ { 2} \)-感觉。到目前为止,尚未研究本文中考虑的阻尼项,并且在低频参数中,该阻尼的效果比经过深入研究的功率类型之一(\((-\ Delta)^ {\ theta} u_ {t} \)与\(\ theta \ in(0,1)\)。当获得最佳衰减率时,我们会遇到带有特殊参数的所谓超几何函数,因此分析似乎更加困难且有吸引力。