The damped wave equation with unbounded damping
Copyright policy )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 turns out to be a robust effect that cannot be easily canceled by adding a positive potential. These analytic results are illustrated by examples.
26 pages, 2 figures
- University of Lisbon Portugal
- Queen's University Belfast United Kingdom
- University of Bern Switzerland
- Instituto Superior de Espinho Portugal
Unbounded damping, quadratic operator function, FOS: Physical sciences, /dk/atira/pure/subjectarea/asjc/2600/2603; name=Analysis, unbounded coeffiecients, 510, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Schrödinger operators with complex potentials, FOS: Mathematics, /dk/atira/pure/subjectarea/asjc/2600/2603, Essential spectrum, unbounded domain, Spectral Theory (math.SP), Mathematical Physics, essential spectrum, One-parameter semigroups and linear evolution equations, name=Analysis, General topics in linear spectral theory for PDEs, Initial value problems for second-order hyperbolic equations, Mathematical Physics (math-ph), Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), Functional Analysis (math.FA), Mathematics - Functional Analysis, 35L05, 35P05, 47A56, 47D06, Quadratic operator function with unbounded coefficients, Damped wave equation, Analysis of PDEs (math.AP)
Unbounded damping, quadratic operator function, FOS: Physical sciences, /dk/atira/pure/subjectarea/asjc/2600/2603; name=Analysis, unbounded coeffiecients, 510, Mathematics - Spectral Theory, Mathematics - Analysis of PDEs, Schrödinger operators with complex potentials, FOS: Mathematics, /dk/atira/pure/subjectarea/asjc/2600/2603, Essential spectrum, unbounded domain, Spectral Theory (math.SP), Mathematical Physics, essential spectrum, One-parameter semigroups and linear evolution equations, name=Analysis, General topics in linear spectral theory for PDEs, Initial value problems for second-order hyperbolic equations, Mathematical Physics (math-ph), Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones), Functional Analysis (math.FA), Mathematics - Functional Analysis, 35L05, 35P05, 47A56, 47D06, Quadratic operator function with unbounded coefficients, Damped wave equation, Analysis of PDEs (math.AP)
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