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Mathematics
Volume 14
Issue 12
10.3390/math14122120
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Article

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

by
Binwei Xie
†,
Syed Omar Shah
† and
Jun Zheng
*
School of Mathematics, Southwest Jiaotong University, Chengdu 611756, China
*
Author to whom correspondence should be addressed.
†
These authors contributed equally to this work.
Mathematics 2026, 14(12), 2120; https://doi.org/10.3390/math14122120 (registering DOI)
Submission received: 30 April 2026 / Revised: 30 May 2026 / Accepted: 10 June 2026 / Published: 14 June 2026
(This article belongs to the Special Issue Stability and Stabilization of Partial Differential Equations)
Versions Notes

Abstract

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.
Keywords: input-to-state stability; coupled PDEs; parabolic equations; generalized Lyapunov method; Dirichlet boundary disturbances input-to-state stability; coupled PDEs; parabolic equations; generalized Lyapunov method; Dirichlet boundary disturbances

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MDPI and ACS Style

Xie, B.; Shah, S.O.; Zheng, J. ISS in Different Norms of Coupled Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances. Mathematics 2026, 14, 2120. https://doi.org/10.3390/math14122120

AMA Style

Xie B, Shah SO, Zheng J. ISS in Different Norms of Coupled Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances. Mathematics. 2026; 14(12):2120. https://doi.org/10.3390/math14122120

Chicago/Turabian Style

Xie, Binwei, Syed Omar Shah, and Jun Zheng. 2026. "ISS in Different Norms of Coupled Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances" Mathematics 14, no. 12: 2120. https://doi.org/10.3390/math14122120

APA Style

Xie, B., Shah, S. O., & Zheng, J. (2026). ISS in Different Norms of Coupled Nonlinear Parabolic PDEs with Dirichlet Boundary Disturbances. Mathematics, 14(12), 2120. https://doi.org/10.3390/math14122120

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