Speaker
Description
We consider the abstract evolution problem $u'(t)=Au(t)+f(t)$ in a
Hilbert space, where $A$ generates a strongly stable but not uniformly
stable $C_0$-semigroup $S(t)$, the typical situation for partially
dissipative systems such as coupled heat--wave models, which arise as
simplifications of fluid--structure interaction. In this setting
classical resonance, i.e.\ an eigenvalue of $A$ on the imaginary axis,
cannot occur. Nevertheless, time-periodic forcing $f$ may fail to
produce a bounded, or any, time-periodic response. We discuss several
notions of resonance appropriate to this setting, organized as a
hierarchy of failures of the range condition $\mathcal R(I-S(T))=H$,
and their connection to the growth of the resolvent of $A$ along the
imaginary axis. As the main example, we show that a heat--wave system
whose wave component occupies an elliptic domain admits periods $T$
with infinite regularity loss between forcing and solution, so that
resonance occurs already for smooth time-periodic forces.