Stabilization of the linear Kawahara equation with localized damping

Abstract

In this erratum of the article: “Stabilization of the linear Kawahara equation with localized damping”, published in Asymptotic Analysis 58(4) (2008), 229–252, DOI 10.3233ASY-2008-0895, we shall fix a mistake which was done when we proved the existence and we defined the countable set of the lengths L where the decay of the energy fails, a set named 𝒩, see Lemma 2.1, p. 235. We considered the equation defined by λu₀+u₀‴+u₀'''''=0, which is wrong since the parameter η in the Kawahara equation was completely forgotten and it makes changing in the result. In fact, η is a negative constant and the correct equation is λu₀+u₀‴+ηu₀'''''=0. Then, as η<0 we can prove, in this case, that the set 𝒩 is empty.

Therefore, we obtain that the energy associated with linear Kawahara equation decays exponentially for all lengths L, even in absence of the damping.

In the new proof for Lemma 2.1, remain the same idea and the same arguments of the original proof, but the signal of parameter η produces different final result.