Carlos Frederico Vasconcellos

Stabilization of the linear Kawahara equation with localized damping

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 + u ′ ′ ′ 0 + u ′ ′ ′ ′ ′ 0 = 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 + u ′ ′ ′ 0 + ηu ′ ′ ′ ′ ′ 0 = 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. 1. New proof for Lemma 2.1 Lemma 1.1. Let N be the set of the values L of the length of interval which satisfies the following conditions: There exist λ ∈ C and a non-trivial u0 ∈ H3 0 (0, L) ∩ H5(0, L) solution of the equation: λu0 + u′ ′ ′ 0 + ηu ′ ′ ′ ′ ′ 0 = 0. (1.1) If η is a negative constant, then N is an empty set. Proof. We divide the proof into two steps: Step 1. In this step we follow closely to the idea developed by Rosier [2], Lemma 3.5. Consider v defined by v(x) = { v0(x) if x ∈ [0, L], 0 if x ∈ R\[0, L]. It is easy to see that v0 satisfies (1.1) if and only if v belongs to H3(R) and satisfies the following equation in D ′(R): λv + v′ ′ ′ + ηv′ ′ ′ ′ ′ = ηv′ ′ ′ 0 (0)δ ′ 0 − ηv′ ′ ′ 0 (L)δ′ L + ηv′ ′ ′ ′ 0 (0)δ0 − ηv′ ′ ′ ′ 0 (L)δL, 0921-7134/10/

The Linear Kawahara Equation on the Half-Line

27.50

Critical set of the Kawahara equation

We study the stabilization of global solutions of the Linear Kawahara (K) equation posed on the right half-line, under the effect of a localized damping mechanism. In this work we analyze the existence, uniqueness and regularity of solutions for the (K) equation, using semigroups theory and since this system is defined on an unbounded domain, a special multiplier argument is showed. To prove the exponential decay of the energy associated to (K) system, due to a lack of compactness, we use local compactness arguments and multipliers techniques. The Kawahara equation describes the evolution of small amplitude long waves in problems arising in fluid dynamics.

xact Controllability and Stabilization for Kuramoto-Sivashinsky System

We characterize the lengths of intervals for which the linear Kawahara equation has a non-trivial solution, whose energy is stationary. This gives rise to a family of complex functions. Characterizing the lengths amounts to deciding which members of this family are entire functions. Our approach is essentially based on determining the existence of certain Mobius transformation.

Stabilization of the linear Kawahara equation

We analyze the stabilization and internal exact control for the Kuramoto- Sivashinsky equation (KS) in a bounded interval. [...]

The stabilization of the nonlinear model arising in pulse propagation in optical fiber

We study the stabilization of global solutions of the linear Kawahara equation (K) with periodic boundary conditions over the interval (0 , 2 π ) under the effect of a localized dam- ping mechanism. The Kawahara equation is a model for small amplitude long waves. Using separation of variables, the Ingham inequality, multiplier techniques and compactness arguments we prove the exponential decay of the solutions of the (K) model.

Equações Integro-Diferenciais do Tipo Barbashin

We analyze the stabilization of the third order nonlinear Schr¨odinger equation in a bounded interval under the effect of a localized damping mechanism.

Stabilization of the Kawahara equation with localized damping

Analisamos equacoes integro-diferenciais do tipo Barbashin quando os multiplicadores c = c ( t,s ) e nucleos k = k ( t,s,μ ) sao estacionarios. Consideramos os casos analisamos o problema quando o multiplicador c e limitado e quando nao e limitado. A existencia e unicidade de solucoes sao deduzidas atraves de teoremas de ponto-fixo.