Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 207

onto L 2 ./, while, by Rellich theorem,

Clearly, A is a homeomorphism of X K

is compact. As a result, is Fredholm of index 0, and the desired existence

result follows if we show that the only solution w n 2 X

A C K

corresponding to F n 0, is
trivial. To this end, we multiply both sides of ( 3.17 ) 1 , with f n 0,by{.LJ=Ǜ/n! u n
(“ ”= c.c.) and integrate by parts over to obtain, also with the help of ( 3.18 )

3 ! 3 LJ

2 C {n! LJ

2 2

2 C {n!LJ. n;x ; u n / D 0:

{ jnj

Ǜ k u n k Ǜ k u n;x k

(3.20)

Likewise, taking the c.c. of ( 3.17 ) 2 , with Q n 0, and then multiplying both sides
by n and integrating over allows us to deduce

2 2

{n!k n k

2 Ck n;x k 2 {n!LJ. n;x ; u n / D 0:

(3.21)

Summing side by side ( 3.20 )and( 3.21 ) then implies k n;x k 2 D 0, that is, in view
of ( 3.18 ), n 0. As a consequence, from ( 3.17 ) 2 (with Q n 0) and again ( 3.18 )
we show, since LJ ¤ 0,that u n must identically vanish, and the claimed existence
result follows. The last statement in the proposition is a consequence of Banach
closed range theorem.

t

With the help of Proposition 3.2 we are now in a position to prove the following
theorem which furnishes the non-occurrence of resonance for a sufficiently large
class of data.

Theorem 3.1. Let f;Q be time-periodic functions of arbitrarily fixed period T>0
with

f 2 W 3;2 .0;T I L 2 .//; Q 2 W 2;2 .0;T I L 2 .//:

Then, there exists one and only one time-periodic solution . u ;/ to ( 3.16 ) of period
T such that

u 2 W 2;2 .0;T I L 2 .// \ W 1;2 .0;T I H 0 ./ \ L 2 .0;T I H 2 .//
2 W 2;2 .0;T I H 0 .// \ W 1;2 .0;T I H 2 .//

Proof. Our starting point is the system of equations ( 3.22 )for arbitrary n 2 Z ,

whose corresponding solutions are provided in Proposition 3.2 . The first objective
is then to show estimates of the type ( 3.19 ) but with an explicit dependence of
the involved constant on the number n. Clearly, in view of ( 3.19 ), it is enough to
establish such estimates for all jnj1. For simplicity, throughout the proof we
shall omit the subscript “n.” Proceeding in the same way as we did to deduce ( 3.20 )
and ( 3.21 ) (this time with f;Q 6 0) we obtain

n 2 ! 2

2 2

k u k

2 Ck u x k 2 C Ǜ. x ; u / D .f ; u /;

(3.22)

2 Ck x k 2 {n!LJ. x ; u / D .Q;/:

{n!kk