Hyperbolic–Parabolic Coupling and the Occurrence of Resonance in Partially Dissipative Systems - Fluid-Structure Interaction and Biomedical Applications - page 221

namely, for any f in the dense set Q there exists x 2 X such that

Z T

.I U/xD U.T s/f.s/ds;

0

and this completes the proof of the second statement in (ii).
We finally prove property (iii). We recall that, for a given T -periodic function f ,
the existence of a corresponding T -periodic solution to ( 3.49 )is equivalent to find
x 2 X satisfying ( 3.55 ). Now, under the assumption ( 3.51 ), by the Banach closed
range theorem .I U/is continuously invertible. Therefore, from ( 3.55 ) we obtain
x D .I U/ 1 Z T
0

U.T s/f.s/ds;

which provides the desired solution. Finally, the corresponding uniqueness result is
an immediate consequence of the first condition in ( 3.51 ). t

Remark 3.6. As immediate consequence of Theorem 3.3 , we deduce the following
result. Necessary and sufficient condition in order that for any T -periodic f 2
L 1 .0;T I X/ Eq. ( 3.49 ) admits a unique T -periodic mild solution is that both
conditions in ( 3.51 ) hold. This extends the analogous result of [ 41 , Theorem 1]
to the case of more general f .

Remark 3.7. If f is such that
Z T

U.T s/f.s/ds D 0;

0

which, of course, is a non-generic property of the data, then the assumption on the
closedness of the range in ( 3.51 ) is not required (since always 0 2 R .I U/)and

the unique T -periodic solution is given by

Z t

x.t/ D U.t s/f.s/ds;

0

see [ 36 , Theorem 3.3].

Remark 3.8. The first part of Theorem 3.3 can be equivalently restated by saying
that if the homogeneous equation (namely ( 3.49 ) with f 0) has a non-
trivial T -periodic solution, then T is a resonant period for the non-homogeneous
equation ( 3.49 ). This is in complete agreement with what we have shown in Sect. 3.3
in the particular case of thermoelasticity with periodic boundary conditions.

Our next objective is to give sufficient conditions on the semigroup U.t/ that
ensure the validity of ( 3.51 )forallT>0. As a by-product, we will then obtain
existence of T -periodic solutions for arbitrary period, which in turn will rule out the
occurrence of resonance. The above conditions are, as somewhat expected, related
to the asymptotic behavior of U.t/ for large t.