Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 202

following one

u tt u . C /rdiv u C Ǜ u t D f in .0; 1/; (3.8)

where Ǜ>0. 5 Therefore, if we repeat step by step the previous procedure, we obtain
that the function w is, this time, a solution to the following equation

! 2 w C {!Ǜ w C L . w / D g:

As a result, we prove

u .x;t/ D e {!t X
k D 1

g k
k C {Ǜ! ! 2 k ;

implying

2 D 2 X
k D 1

2

jg k j

k u .t/k

C Ǜ 2 ! 2 2 ; Œ. k ! 2 / 2

from which we deduce at once that resonance does not occur.
Another point that is worth remarking and that will be useful for future
considerations regards the behavior in time of the total energy E :

2 k u t .t/k

2 ; 1 2 C kr u k 2 C . C /kdiv u k

E .t/ WD

in absence of external loads. Actually, in the purely elastic case described by ( 3.2 ),
E is a constant function of time (conservation of energy):

E .t/ D E .0/; for all t 0: (3.9)

This property is well known and easy to show by dot-multiplying both sides of ( 3.2 )
with f 0 by u andintegratingbypartsover.
In the presence of linear damping, however, the total energy decays exponentially
fast , a property that, as we shall see further on, can be intimately related to the lack
of resonant effects. In order to sketch the proof of the above decay, we dot-multiply
both sides of ( 3.8 ) with f 0,by u t and u , respectively, and integrate by parts over
to obtain

d dt DǛk u t k

2
2

h 2 k u k

2 C . u t ; u / i

d
dt

2 2 2 2

Dk u t k

2 kr u k 2 . C /kdiv u k 2 :

5 The calculations to follow show lack of resonance also in the case Ǜ<0.However,suchan
assumption is unacceptable from the physical viewpoint in that it would imply an increase of total
energy in absence of external loads.