  This work provides stability results in the spatial sup norm for
hyperbolic-parabolic loops in one spatial dimension. The results are obtained
by an application of the small-gain stability analysis. Two particular cases
are selected for the study because they contain challenges typical of more
general systems (to which the results are easily generalizable but at the
expense of less pedagogical clarity and more notational clutter): (i) the
feedback interconnection of a parabolic PDE with a first-order zero-speed
hyperbolic PDE with boundary disturbances, and (ii) the feedback
interconnection, by means of a combination of boundary and in-domain terms, of
a parabolic PDE with a first-order hyperbolic PDE. The first case arises in the
study of the movement of chemicals underground and includes the wave equation
with Kelvin-Voigt damping as a subcase. The second case arises when we apply
backstepping to a pair of hyperbolic PDEs that is obtained by ignoring
diffusion phenomena. Moreover, the second case arises in the study of parabolic
PDEs with distributed delays. In the first case, we provide sufficient
conditions for ISS in the spatial sup norm with respect to boundary
disturbances. In the second case, we provide (delay-independent) sufficient
conditions for exponential stability in the spatial sup norm.
