Decay estimates for 1-D parabolic PDES with boundary disturbances

¹ Department of Mathematics, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece
² Department of Mechanical and Aerospace Engineering, University of California, San Diego, La Jolla, CA 92093-0411, USA

^* Corresponding author: iasonkar@central.ntua.gr

Received: 4 June 2017
Accepted: 18 July 2018

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.