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In this paper, we introduce the notion of relative$\mathcal{K}$-equi-stability (RKES) to characterize the uniformly continuousdependence of (weak) solutions on external disturbances for nonlinear parabolicPDE systems. Based on the RKES, we prove the input-to-state stability (ISS) inthe spatial sup-norm for a class of nonlinear parabolic PDEs with eitherDirichlet or Robin boundary disturbances. Two examples, concerned respectivelywith a super-linear parabolic PDE with Robin boundary condition and a $1$-Dparabolic PDE with a destabilizing term, are provided to illustrate theobtained ISS results. Besides, as an application of the notion of RKES, weconduct stability analysis for a class of parabolic PDEs in cascade coupledover the domain or on the boundary of the domain, in the spatial and timesup-norm, and in the spatial sup-norm, respectively. The technique of De Giorgiiteration is extensively used in the proof of the results presented in thispaper.

Relative Stability in the Sup-norm and Input-to-state Stability in the Spatial Sup-norm for Parabolic PDEs