论文标题
在具有可变系数的一维波方程的指数级时间上
On the exponential time-decay for the one-dimensional wave equation with variable coefficients
论文作者
论文摘要
我们考虑了具有正时lipschitz连续系数的一维，时间依赖性波方程的初始问题问题，它们是在有限区域之外恒定的。在对初始数据的紧凑支持的假设下，我们证明了局部能量的时间呈指数速度，并提供了解决方案在大时收敛的明确常数。我们通过两种不同的技术对这种指数衰减的速率进行明确估计。第一个是基于适当构造的重量的修改后的加权局部能量的定义。第二个是基于问题的积分公式，并且在对系数变化的更限制的假设下，使我们能够获得改善的衰减率。
We consider the initial-value problem for the one-dimensional, time-dependent wave equation with positive, Lipschitz continuous coefficients, which are constant outside a bounded region. Under the assumption of compact support of the initial data, we prove that the local energy decays exponentially fast in time, and provide the explicit constant to which the solution converges for large times. We give explicit estimates of the rate of this exponential decay by two different techniques. The first one is based on the definition of a modified, weighted local energy, with suitably constructed weights. The second one is based on the integral formulation of the problem and, under a more restrictive assumption on the variation of the coefficients, allows us to obtain improved decay rates.
