  In this paper, we introduce the notion of relative
$\mathcal{K}$-equi-stability (RKES) to characterize the uniformly continuous
dependence of (weak) solutions on external disturbances for nonlinear parabolic
PDE systems. Based on the RKES, we prove the input-to-state stability (ISS) in
the spatial sup-norm for a class of nonlinear parabolic PDEs with either
Dirichlet or Robin boundary disturbances. Two examples, concerned respectively
with a super-linear parabolic PDE with Robin boundary condition and a $1$-D
parabolic PDE with a destabilizing term, are provided to illustrate the
obtained ISS results. Besides, as an application of the notion of RKES, we
conduct stability analysis for a class of parabolic PDEs in cascade coupled
over the domain or on the boundary of the domain, in the spatial and time
sup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgi
iteration is extensively used in the proof of the results presented in this
paper.
