Wave Equation for Operators with Discrete Spectrum and Irregular Propagation Speed

Abstract

Given a Hilbert space H, we investigate the well-posedness of the Cauchy problem for the wave equation for operators with a discrete non-negative spectrum acting on 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 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.

Bitsadze–Samarskii type problem for the integro-differential diffusion–wave equation on the Heisenberg group

“…We begin with a reminder of the definition of the group Fourier transform on the Heisenberg group (see many sources, but e.g. [26,27] for its use in similar contexts). For f ∈ S(H n ) the group Fourier transform is defined as…”

Spectral properties of the iterated Laplacian with a potential in a punctured domain

“…One can be started a 'nonharmonic' analysis connected with the singular, in the above sense, operators. Note, that the nonharmonic analysis is developed in the works [2,10,13,15] with applications given in [14,16]. For more general setting of the nonharmonic analysis, see for instance [6].…”

On numerical simulations of the 1d wave equation with a distributional coefficient and source term

“…From [4] it follows that the problem (1) has a unique very weak solution. It is given by a family of functions �� � (�, �)� ����� .…”

“…In this paper, we follow the results of the paper [4] and study the Cauchy-Dirichlet problem for the 1D-Wave Equation The notion of very weak solutions has been introduced in [GR15] to analyse second order hyperbolic equations. In [3] and [5] Ruzhansky and Tokmagambetov applied it to show the wellposedness of the Landau Hamiltonian wave equations in distributional electro-magnetic fields.…”