Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 206

appropriate variables, we then obtain, with the obvious meaning of the symbols,
that the relevant equations take the following form

u tt u xx C Ǜ x D f
t xx C LJ u xt D Q
u .0;t/ D u .`;t/ D .0;t/ D .`;t/ D 0t>0;

(3.16)

where ǛLJ > 0. We next expand u ;;f,andQ in Fourier series in time with period
T D 2=!, namely

u .x;t/ D X
n 2Z

u n .x/e {n!t ; etc.

From ( 3.16 ) we then infer that the corresponding coefficients u n ; n ;f n ,andQ n
must satisfy the following equations

0<x<`;

n 2 ! 2 u n u n;xx C Ǜ n;x D f n .x/
{n! n n;xx C {n!LJ u n;x D Q n .x/

(3.17)

with boundary conditions

u n .0/ D u n .`/ D n .0/ D n .`/ D 0: (3.18)

Our next result ensures the existence of solutions to ( 3.17 )-( 3.18 ) with corre-
sponding estimates.

2 L 2 ./ , n 2 Z

Proposition 3.2. For any f n ;Q n , there exists one and only

one solution . u n ; n / to ( 3.17 ) - ( 3.18 ) such that . u n ; n / 2 ŒH 2 ./ \ H 0 ./ 2 .
Furthermore, there exists C D C.n;!;`;Ǜ;LJ/>0 such that
k u n k 2;2 Ck n k 2;2 C kf n k 2 CkQ n k 2 : (3.19)

Proof. Set (D WD d=dx)

D 2

D 2 ;

n 2 ! 2 ǛD
{n!LJD{n!

;

0

A D K D

0

u n
n

;F n D f n

Q n

:

w n D

X WD ŒH 2 ./\H 0 ./ 2 in L 2 ./,

The operators A and K are well defined from

so that for each n 2 Z our problem ( 3.17 )-( 3.18 ) can be written in the following

abstract form

. w n / D F n in L 2 ./: A . w n / C K