ISS in Different Norms of Coupled Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances

This paper addresses the input-to-state stability (ISS) in different Lq-norms for a class of coupled nonlinear partial differential equations of parabolic type subject to both in-domain disturbances and Dirichlet boundary disturbances, where q∈[1,+∞). Specifically, we first prove the continuous dependence of solutions to the system on initial data and disturbances in different Lq-norms by using the generalized Lyapunov method, and subsequently derive ISS estimates via a density argument. The main challenge arises in handling the nonlinear coupling terms and deriving ISS small-gain conditions within the generalized Lyapunov framework, as each coupling term depends on all other state variables of the system.