  A paper, entitled "Uniform stabilization for the Timoshenko beam by a locally
distributed damping" was published in 2003, in the journal Electronic Journal
of Differential Equations. Its title concerns exclusively its Section 3,
devoted to the case of equal speeds of propagation and to its main theorem,
namely Theorem 3.1. It states that the solutions of the Timoshenko system (see
(1.3) in [1]) decays exponentially when the damping coefficient b is locally
distributed. The proof of Theorem 3.1 is crucially based on Lemma 3.6, which
states the existence of a strict Lyapunov function along which the solutions of
(1.3) decay when the speeds of propagation are equal. This rebuttal shows the
major gap and flaws in the proof of Lemma 3.6, which invalidate the proofs of
Lemma 3.6 and Theorem 3.1. Lemma 3.6 is stated at the top of page 12. The main
part of its proof is given in the pages 12 and 13. In the last eight lines of
page 13, eight inequalities are requested to hold together for the proof of
Lemma 3.6. They don't appear in the statements of Lemma 3.6. The subsequent
flaws come from the evidence that several of them are contradictory either
between them or with claims in the title of the article. We also point in this
rebuttal other flaws, or gaps in the proofs of Theorem 2.2 related to strong
stability and non uniform stability for the case of distinct speeds of
propagation. In [3], we correct and complete the proof of strong stability. We
also correct, set up the missing functional frames, fill the gaps in the proof
of non uniform stability in the cases of different speeds of propagation, and
complete a missing argument in the proof of Theorem A in [4] (see Remark 4.3),
the result of Theorem A being used in the paper [1] on which this rebuttal is
mainly devoted.
