论文标题
具有奇异时间依赖性传播速度的抽象波方程的有限与无限导数损失
Finite vs infinite derivative loss for abstract wave equations with singular time-dependent propagation speed
论文作者
论文摘要
我们考虑一个仅取决于时间的传播速度的抽象波动方程。在传播速度平稳的情况下，在正时，但在初始时间可能是单数的，我们研究了适合的结果，并具有有限的导数损失。 我们证明，解决方案在涉及传播速度的第一和第二个衍生物的爆炸率下表现出有限的导数损失，因为较弱的是对第一个衍生物的要求，因此较弱的是对第二个导数的要求。我们的条件家庭在文献中已经知道的两个限制案例之间进行了插值。 我们还提供反例表明，一旦我们的条件失败，解决方案就会表现出无限的导数损失。即使在两种极端情况下，这种病理的存在也是一个空旷的问题。
We consider an abstract wave equation with a propagation speed that depends only on time. We investigate well-posedness results with finite derivative loss in the case where the propagation speed is smooth for positive times, but potentially singular at the initial time. We prove that solutions exhibit a finite derivative loss under a family of conditions that involve the blow up rate of the first and second derivative of the propagation speed, in the spirit that the weaker is the requirement on the first derivative, the stronger is the requirement on the second derivative. Our family of conditions interpolates between the two limit cases that were already known in the literature. We also provide the counterexamples that show that, as soon as our conditions fail, solutions can exhibit an infinite derivative loss. The existence of such pathologies was an open problem even in the two extreme cases.
