论文标题
各向异性阻尼波的急剧指数衰减速率
Sharp exponential decay rates for anisotropically damped waves
论文作者
论文摘要
在本文中，我们研究了紧凑型riemannian歧管上阻尼波方程的能量衰减，其中阻尼系数是各向异性的，并由零阶的伪差算子建模。我们证明，当且仅当阻尼系数满足经典几何控制条件的各向异性类似以及独特的持续假设时，溶液的能量就以指数的速率衰减。此外，我们根据光谱横坐标和最佳衰减速率的明确公式，以及与大地测量学上阻尼的主要符号的长期平均值，类似于列博对各向同性案例的工作。我们还构建了真正的各向异性阻尼，以满足我们在平坦的圆环上的假设。
In this article, we study energy decay of the damped wave equation on compact Riemannian manifolds where the damping coefficient is anisotropic and modeled by a pseudodifferential operator of order zero. We prove that the energy of solutions decays at an exponential rate if and only if the damping coefficient satisfies an anisotropic analogue of the classical geometric control condition, along with a unique continuation hypothesis. Furthermore, we compute an explicit formula for the optimal decay rate in terms of the spectral abscissa and the long-time averages of the principal symbol of the damping over geodesics, in analogy to the work of Lebeau for the isotropic case. We also construct genuinely anisotropic dampings which satisfy our hypotheses on the flat torus.
