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Dissipative operators with finite dimensional damping

Published online by Cambridge University Press:  14 November 2011

Harald Röh
Harald Röh
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Currie, Edinburgh EH14 4AS
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Synopsis

Let G: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ and T(t) be the Co semigroup generated by G. A spectral mapping theorem σ(T(t))\{0} = exp (tσ(G))/{0} together with a condition for 0 ε σ(T(t)) are proved if the set {x ε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions for G are satisfied. Proofs are given in terms of the Cayley transformation T = (G + I)(G − I)−1 of G. The results are applied to the damped wave equation utt + γutx + uxxxx + ßuxx = 0, 0 ≦ t < ∞ 0 < x < 1, β, γ ≧ 0, with boundary conditions u(0, t) = ux(0, t) = uxx (1, t) = uxxx(1, t) = 0.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 91 , Issue 3-4 , 1982 , pp. 243 - 263
Copyright
Copyright © Royal Society of Edinburgh 1982

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