Dissipative operators with finite dimensional damping
1982 ◽
Vol 91
(3-4)
◽
pp. 243-263
◽
Keyword(s):
SynopsisLetG: ε(G)⊂ℋ → ℋ be a maximal dissipative operator with compact resolvent on a complex separable Hilbert space ℋ andT(t) be theCosemigroup generated byG. 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 {xε ⅅ(G) | Re (Gx, x) = 0} has finite codimension in ε(G) and if some eigenvalue conditions forGare satisfied. Proofs are given in terms of the Cayley transformationT= (G+I)(G−I)−1ofG. The results are applied to the damped wave equationutt+ γutx+uxxxx+ ßuxx= 0, 0 ≦t< ∞ 0 <x< 1, β, γ ≧ 0, with boundary conditionsu(0,t) =ux(0,t) =uxx(1,t) =uxxx(1,t) = 0.
1996 ◽
Vol 38
(1)
◽
pp. 61-64
◽
2015 ◽
Vol 268
(9)
◽
pp. 2479-2524
◽
1985 ◽
Vol 8
(2)
◽
pp. 276-288
◽
Keyword(s):
Keyword(s):
2003 ◽
Vol 2003
(16)
◽
pp. 933-951
◽
2015 ◽
Vol 423
(1)
◽
pp. 1-9
◽