Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed
 作者： Michael Ruzhansky,  Niyaz Tokmagambetov 作者单位： 1Imperial College London2al–Farabi Kazakh National University 刊名： Archive for Rational Mechanics and Analysis, 2017, Vol.226 (3), pp.1161-1207 来源数据库： Springer Nature Journal DOI： 10.1007/s00205-017-1152-x 英文摘要： Given a Hilbert space $${\mathcal{H}}$$ , we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on $${\mathcal{H}}$$ . We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples... include the harmonic and anharmonic oscillators, the Landau Hamiltonian on $${\mathbb{R}^n}$$ , uniformly elliptic operators of different orders on domains, Hörmander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others. 原始语种摘要： Given a Hilbert space $${\mathcal{H}}$$ , we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on $${\mathcal{H}}$$ . We consider the cases when the time-dependent propagation speed is regular, Hölder, and distributional. We also consider cases when it is strictly positive (strictly hyperbolic case) and when it is non-negative (weakly hyperbolic case). When the propagation speed is a distribution, we introduce the notion of “very weak solutions” to the Cauchy problem. We show that the Cauchy problem for the wave equation with the distributional coefficient has a unique “very weak solution” in an appropriate sense, which coincides with classical or distributional solutions when the latter exist. Examples... include the harmonic and anharmonic oscillators, the Landau Hamiltonian on $${\mathbb{R}^n}$$ , uniformly elliptic operators of different orders on domains, Hörmander’s sums of squares on compact Lie groups and compact manifolds, operators on manifolds with boundary, and many others.

• problem　题目
• consider　仔细考虑
• operators　操作符
• anharmonic　非低的
• strictly　严密地
• Hamiltonian　哈密(尔)顿
• compact　紧的
• propagation　传播
• notion　概念
• dependent　从属的