Relative stability in the sup-norm and input-to-state stability in the spatial sup-norm for parabolic PDEs

In this paper, we introduce the notion of relativeK-equi-stability (RKES) to characterize the uniformlycontinuous dependence of (weak) solutions on externaldisturbances for nonlinear parabolic PDE systems. Basedon the RKES, we prove the input-to-state stability (ISS)in the spatial sup-norm for a class of nonlinear parabolicPDEs with either Dirichlet or Robin boundary disturbances.An example concerned with a super-linear parabolic PDEwith Robin boundary condition is provided to illustratethe obtained ISS results. Besides, as an application of thenotion of RKES, we conduct stability analysis for a classof parabolic PDEs in cascade coupled over the domain oron the boundary of the domain, in the spatial and timesup-norm, and in the spatial sup-norm, respectively. Thetechnique of De Giorgi iteration is extensively used in theproof of the results presented in this paper.

Uncontrolled Keywords

Nonlinear PDEs, relative stability, input-tostate stability, De Giorgi iteration, cascade of PDE systems.