Decay Estimates for 1-D Parabolic PDEs with Boundary Disturbances

TL;DR

This paper derives decay estimates in L2 and H1 norms for solutions of 1-D linear parabolic PDEs with boundary and distributed disturbances, applicable even with discontinuous disturbances, aiding stability analysis.

Contribution

It provides decay estimates that do not require eigenvalues or eigenfunctions, extending stability analysis methods for parabolic PDEs with boundary disturbances.

Findings

Decay estimates in L2 and H1 norms established

Applicable to discontinuous disturbances

Useful for stability analysis of PDEs with nonlocal terms

Abstract

In this work decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the L2 and H1 norms of the solution and discontinuous disturbances are allowed. Although an eigenfunction expansion for the solution is exploited for the proof of the decay estimates, the estimates do not require knowledge of the eigenvalues and the eigenfunctions of the corresponding Sturm-Liouville operator. Examples show that the obtained results can be applied for the stability analysis of parabolic PDEs with nonlocal terms.