Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 222

To this end, we recall that a semigroup S.t/ is called strongly stable if

t !1 kS.t/xkD0; for all x 2 X ,

lim (3.63)

whereas it is called uniformly stable if

t !1 kS.t/kD0:

lim (3.64)

The following result is well known; see, e.g., [ 5 , Remark at p. 178].

Lemma 3.1. A strongly continuous semigroup S.t/ is uniformly stable if and only
if there exists M;ı > 0 such that

kS.t/xkM e ıt kxk ; for all t 0 and all x 2 X . (3.65)

The next theorem shows that strong stability prevents resonance for dense set of
data but does not rule it out completely.

Theorem 3.4. Suppose that U.t/ is strongly stable, and let T>0 be arbitrarily
given. Then we can find a dense set

Q L 1 .0;T I X/ such that for any T -periodic

f 2 Q there exists a unique corresponding T -periodic solution x 2 C.Œ0;TI X/

to Eq. ( 3.49 ).

Proof. By Theorem 3.3 (ii) it is enough to show

N .I U.T// Df0g for all T>0. (3.66)

Assume ( 3.63 ) holds and fix T>0. Then, for any ">0there is n 2 N such that

kU.nT/xk <";for all n n: (3.67)

We n ow h ave , f o r a l l n>n,

" U.T/x C U.2T/x CCU.nT/x C

U.kT/x # :

n

X

1
n

U n .T/x D

k D n C 1

Thus, using the property (iv) of the semigroup and ( 3.67 )

1
n Œn C ".n n/kxk ;

kU n .T/xk

from which, by the arbitrariness of ", we conclude

n !1 kU n .T/xkD0:

lim

Therefore ( 3.66 ) follows.