The damped wave equation with singular damping

具有无界阻尼的阻尼波方程的伪谱

Pseudospectra of the damped wave equation with unbounded damping

本文分析了具有无界阻尼的阻尼波方程生成元的伪谱，证明预解式范数发散及算子非正规特征的稳健性，揭示了谱不稳定性及相关伪谱效应的存在。

The damped wave equation with unbounded damping

We analyze new phenomena arising in linear damped wave equations on unbounded domains when the damping is allowed to become unbounded at infinity. We prove the generation of a contraction semigroup, study the relation between the spectra of the semigroup generator and the associated quadratic operator function, the convergence of non-real eigenvalues in the asymptotic regime of diverging damping on a subdomain, and we investigate the appearance of essential spectrum on the negative real axis. We further show that the presence of the latter prevents exponential estimates for the semigroup and t…

Sharp polynomial decay rates for the damped wave equation with Hölder-like damping

We study decay rates for the energy of solutions of the damped wave equation on the torus. We consider dampings invariant in one direction and bounded above and below by multiples of $x^β$ near the boundary of the support and show decay at rate $1/t^{\frac{β+2}{β+3}}$. In the case where $W$ vanishes exactly like $x^β$ this result is optimal by work of the second author. The proof uses a version of the Morawetz multiplier method.

From Lieb-Thirring inequalities to spectral enclosures for the damped wave equation

Using a correspondence between the spectrum of the damped wave equation and non-self-adjoint Schroedinger operators, we derive various bounds on complex eigenvalues of the former. In particular, we establish a sharp result that the one-dimensional damped wave operator is similar to the undamped one provided that the L^1 norm of the (possibly complex-valued) damping is less than 2. It follows that these small dampings are spectrally undetectable.

Eigenvalue asymptotics, inverse problems and a trace formula for the linear damped wave equation

We determine the general form of the asymptotics for Dirichlet eigenvalues of the one-dimensional linear damped wave operator. As a consequence, we obtain that given a spectrum corresponding to a constant damping term this determines the damping term in a unique fashion. We also derive a trace formula for this problem.

通过非多项式导数界条件获得阻尼波方程的精确能量衰减率

Sharp energy decay rates for the damped wave equation on the torus via non-polynomial derivative bound conditions

本文研究了在紧致流形上阻尼波方程的能量衰减率，通过非多项式导数界条件获得更一般的衰减率，并展示了这些速率如何依赖于阻尼支持集的几何特性。