Part II On strong and non uniform stability of locally damped Timoshenko beam: Mathematical corrections to the proof of Theorem 2.2 in the publication referenced as [1] in the bibliography

TL;DR

This paper corrects and completes the mathematical proofs related to the stability of Timoshenko beams with local damping, addressing previous gaps and errors in the literature.

Contribution

It provides precise functional frameworks, fills gaps, and corrects proofs of stability theorems for Timoshenko beams, including new propositions in Banach spaces.

Findings

Corrected proofs of strong and non-uniform stability for Timoshenko beams.

Established accurate functional frames for stability analysis.

Filled gaps and fixed errors in previous stability proofs.

Abstract

In part I of the rebuttal (see [2] to the article [1] entitled "Uniform stabilization for the Timoshenko beam by a locally distributed damping" published in 2003, in the journal Electronic Journal of Differential Equations, we prove that Lemma 3.6 and Theorem 3.1 are unproved due to major flaws (contradictory assumptions). We also show that Theorem 2.2 and its proofs of strong stability, and non uniform stability in the case of different speeds of propagation, contain several incorrect arguments and several gaps (including missing functional frames). In this part II, we give the precise missing functional frames, fill the gaps and correct several parts contained in the proof of Theorem 2.2 in [1]. We also complete a missing argument (see Remark 4.23 and Remark 3.2) in the proof of Theorem A in [5] used by [1]. For this we state and prove Proposition 4.4 (see also Proposition 4.6 for a general formulation in Banach spaces). We also give the correct formulations, and proofs of strong stability and non uniform stability (in case of different speeds of propagation) for Timoshenko beams.