Pseudospectra of the Damped Wave Equation with Unbounded Damping

Abstract

We analyze pseudospectra of the generator of the damped wave equation with unbounded damping. We show that the resolvent norm diverges as Re z → −∞. The highly non-normal character of the operator is a robust effect preserved even when a strong potential is added. Consequently, spectral instabilities and other related pseudospectral effects are present.

Resolvent estimates for one-dimensional Schrödinger operators with complex potentials

“…4.2]. We also show the behaviour of Ψ(λ) for the operator with sub-linear potential V (x) = i x 2 3 in Fig. 7.1, remarking that the completeness of the eigensystem for this operator with Dirichlet boundary conditions in L 2 (R + ) was proved in [33].…”

“…[13], where lower estimates for the resolvent norm inside the numerical range of H, Num(H), were obtained by a semi-classical pseudomode construction. The latter was subsequently generalised: in the semi-classical case in particular in [35,16] and in the non-semi-classical one in [26,3,25,18].…”

Semi-uniform stability of operator semigroups and energy decay of damped waves

“…The classical damped wave equation discussed here is, of course, just one (prototypical) example of how abstract semigroup results may be applied to concrete PDEs. We conclude by mentioning a small selection of other interesting applications of semigroup methods to differential equations, along with sample references: damped wave equations on unbounded domains/manifolds [4,39,51,83,85,104,117]; local energy decay for damped wave equations [31,130]; Klein-Gordon and Kelvin-Vogt type equations [7,36,151]; energy decay for non-linear damped wave equations [24,81,82,119]; vectorial damped wave equations [86]; damped wave equations with unbounded and/or indefinite dampings [1,12,25,61,62]; viscoelastic boundary dampings [140]; wave equations with periodic (or even general non-stationary) dampings [83,96]; fractional damped wave equations [65,66]; damped wave equations on manifolds with rough metrics [143]. Even though the subject of damped wave equations is already vast, we hope and expect that the stream of substantial advances in this area, whether obtained by abstract techniques or otherwise, will continue for many years to come.…”

Resolvent estimates for the one-dimensional damped wave equation with unbounded damping

“…[7,11,27,5]) and non-semi-classical (e.g. [24,22,3,12,4]) methods. One approach, pioneered in [7] and subsequently developed non-semi-classically in [24,3,22,12], relies on the construction of pseudomodes (or approximate eigenfunctions) for the operator at hand (Schrödinger, damped wave equation, Dirac, biharmonic) inside the numerical range thereby finding lower bounds on (H − λ) −1 .…”

“…[24,22,3,12,4]) methods. One approach, pioneered in [7] and subsequently developed non-semi-classically in [24,3,22,12], relies on the construction of pseudomodes (or approximate eigenfunctions) for the operator at hand (Schrödinger, damped wave equation, Dirac, biharmonic) inside the numerical range thereby finding lower bounds on (H − λ) −1 . For Schrödinger operators with complex potentials, lower and upper bounds have recently been found in [4] using different (non-semi-classical) methods.…”