Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 205

( 3.15 ) , 0 is the only solution for all >0 , whenever is such that all eigen-
values of the Laplace operator are simple [ 15 ] , which is indeed true for a residual
set of (bounded) domains of class C 2 , in the sense of the Baire category [ 45 ] . 6

This classical result is of great physical relevance, in that it ensures that, for
“most” (and sufficiently regular) bounded domains the dissipation due to thermal
effects is able to damp out the free vibrations of the elastic body (within the above
model, of course). Its proof can be obtained by LaSalle's invariance principle [ 23 ,
43 ], and is based on the following facts. Let
X WD ǚ . u ; u t ;/2 ŒH 1 ./ 3 L 2 ./ ; ŒL 2 ./ 3

be the “energy space.” The set of cluster points (as t !1) of all trajectories
emanating from some x 2 X is contained in the subset, F ,ofX where the
Liapounov functional is constant. From ( 3.12 )to( 3.14 ) (with f Q 0)

we see that F is the space of solutions . u ;/ . u ;0/such that

E

in .0; 1/;

u tt u . C /r.div u / D 0
div u t D 0

u t D 0 on @:

Setting w WD u t , and expanding w in a time Fourier series, the above system may
be equivalently reduced to ( 3.15 ). Therefore, if satisfies the condition stated in
Proposition 3.1 , every trajectory converges to 0 in the X-norm (equivalent to

1
2 ).
The question that we would like to address next is whether such a dissipative
mechanism is also able to rule out the occurrence of resonance.
In this regard, it must be emphasized that Proposition 3.1 does not guarantee,
in general, any uniform rate of decay for

E

.t/. 7 It should also be remarked that
such a decay is not true if instead of Dirichlet boundary conditions, we use periodic
boundary conditions . 8 We will analyze these questions in Sect. 3.3.2 , and in a much
broader context, in Sect. 3.4 .

E

3.3.1 The One-Dimensional Case

In order to perform our study on resonance, we begin to consider the simple
one-dimensional version of ( 3.12 ), where D .0;`/. By suitably rescaling the

6 It is worth remarking that there are also “familiar” domains where ( 3.15 ) has an infinite number of
linearly independent solutions. This happens, for example, when is a ball [ 15 , Remark 5.2], and,
in fact, in the two-dimensional case, the circle is the only (sufficiently smooth) simply connected
domain where ( 3.15 ) has an infinite number of linearly independent solutions [ 8 ].
7 In this respect, see [ 31 ].
8 See, however, also Remark 3.4 .