Altmetric

Wave equation for operators with discrete spectrum and irregular propagation speed

File Description SizeFormat 
10.1007%2Fs00205-017-1152-x.pdfPublished version670.08 kBAdobe PDFView/Open
Title: Wave equation for operators with discrete spectrum and irregular propagation speed
Authors: Ruzhansky, M
Tokmagambetov, N
Item Type: Journal Article
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 Rn, 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.
Issue Date: 26-Jul-2017
Date of Acceptance: 7-Jul-2017
URI: http://hdl.handle.net/10044/1/50083
DOI: https://dx.doi.org/10.1007/s00205-017-1152-x
ISSN: 1432-0673
Publisher: Springer Verlag (Germany)
Start Page: 1161
End Page: 1207
Journal / Book Title: Archive for Rational Mechanics and Analysis
Volume: 226
Issue: 3
Copyright Statement: © The Author(s) (2017) This article is an open access publication
Sponsor/Funder: Engineering & Physical Science Research Council (EPSRC)
The Leverhulme Trust
Funder's Grant Number: EP/K039407/1
RPG-2014-002
Keywords: Science & Technology
Physical Sciences
Technology
Mathematics, Applied
Mechanics
Mathematics
MAGNETIC SCHRODINGER-OPERATORS
HYPERBOLIC CAUCHY-PROBLEMS
DISCONTINUOUS COEFFICIENTS
TRACE FORMULA
ULTRADISTRIBUTIONS
EIGENFUNCTIONS
PERTURBATIONS
ASYMPTOTICS
REGULARITY
FIELD
math.AP
math-ph
math.MP
35G10
0101 Pure Mathematics
0102 Applied Mathematics
General Physics
Publication Status: Published
Appears in Collections:Pure Mathematics
Mathematics
Faculty of Natural Sciences



Items in DSpace are protected by copyright, with all rights reserved, unless otherwise indicated.

Creative Commonsx