Linear wave equations with time-dependent propagation speed and strong damping
Linear wave equations with time-dependent propagation speed and strong damping
复制标题
具有与时间相关的传播速度和强阻尼的线性波动方程
DOI：
10.1016/j.jde.2015.09.037
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发表时间：
2016
影响因子：
2.4
通讯作者：
Massimo Gobbino
中科院分区：
文献类型：
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作者：
M. Ghisi;Massimo Gobbino
我们考虑一个二阶线性方程，其“弹性”算子前面有一个与时间相关的系数 c (t)。对于这些方程，众所周知，初始数据的较高空间规律性可以补偿 c (t) 的较低时间规律性。在本文中，我们研究了强耗散的影响，即取决于弹性算子功率的摩擦项。我们发现的是阈值效应。当摩擦项中弹性算子的指数大于 1/2 时，阻尼占主导地位，方程的表现就好像系数 c (t) 恒定一样。当指数小于1/2时，c(t)的时间规律性开始发挥作用。如果 c (t) 足够规则，则阻尼再次占上风。相反，当 c (t) 不够规则时，阻尼可能无效，并且存在耗散方程表现为非耗散方程的示例。正如预期的那样，阻尼越强，时间规律性阈值越低。我们还提供了反例来显示我们结果的最优性。
We consider a second order linear equation with a time-dependent coefficient c (t) in front of the “elastic” operator. For these equations it is well-known that a higher space-regularity of initial data compensates a lower time-regularity of c (t). In this paper we investigate the influence of a strong dissipation, namely a friction term which depends on a power of the elastic operator. What we discover is a threshold effect. When the exponent of the elastic operator in the friction term is greater than 1/2, the damping prevails and the equation behaves as if the coefficient c (t) were constant. When the exponent is less than 1/2, the time-regularity of c (t) comes into play. If c (t) is regular enough, once again the damping prevails. On the contrary, when c (t) is not regular enough the damping might be ineffective, and there are examples in which the dissipative equation behaves as the non-dissipative one. As expected, the stronger is the damping, the lower is the time-regularity threshold. We also provide counterexamples showing the optimality of our results.