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Erratum: Stabilization of the linear Kawahara equation with localized damping (Asymptotic Analysis (2008) 58(4) (229-252))
... As far as we know, the first works dealing with these kinds of control problems for Kawahara equation were [39,40,41]. In these works, the stabilization of global solutions of the linear Kawahara equation using localized damping was addressed following the ideas presented in the works [33,30], where similar problems were analyzed for the KdV equation. ...
... Observe that contrary to the case of boundary saturation, we first analyze a system which is linear so that we use linear semigroup theory for the study of (40). In what follows, we add the nonlinear terms as source terms. ...
... In [39,40], it was shown that the set of critical lengths related to this equation is empty. Therefore, we expect that our stability result by saturated boundary feedback is still valid in this both context. ...
... We observe that, according to the above energy dissipation law, even when a ≡ 0, E(t) is a nonincreasing function. The linearized problem, in which the term uu x is missing and the damping a ≡ 0, was analyzed by Vasconcellos and da Silva [25,26]. It has been proved that the linear semigroup decays exponentially for all values of the length of the space-interval L and of η < 0. In [25] this result was proved by excluding a countable set of critical lengths L but later on in [26] it was shown that the same holds for all L, unlike the case of the KdV in which there is effectively a countable set of lengths for which the energy of the corresponding linear system does not decay (see [15,19]). ...
... The linearized problem, in which the term uu x is missing and the damping a ≡ 0, was analyzed by Vasconcellos and da Silva [25,26]. It has been proved that the linear semigroup decays exponentially for all values of the length of the space-interval L and of η < 0. In [25] this result was proved by excluding a countable set of critical lengths L but later on in [26] it was shown that the same holds for all L, unlike the case of the KdV in which there is effectively a countable set of lengths for which the energy of the corresponding linear system does not decay (see [15,19]). In this work we study the nonlinear (K) system. ...
... 1), we know that A generates a strongly continuous semigroup of contractions in L 2 (0, L). Let {S(t)} t≥0 be the semigroup of contractions generated by A, then according to [25] (Thm. 1.1), we know that the semigroup of contractions {S(t)} t≥0 satisfies the following properties: ...
We study the stabilization of global solutions of the
Kawahara (K) equation in a bounded interval, under the effect of
a localized damping mechanism. The Kawahara equation is a model
for small amplitude long waves. Using multiplier techniques and
compactness arguments we prove the
exponential decay of the solutions of the (K) model. The proof
requires of a unique continuation theorem and the smoothing effect
of the (K) equation on the real line, which are proved in this work.
... In [42,44] the damping like in (8) was used for (7) without the drift term u x . If, however, the linear term u x is dropped, both the KdV and Kawahara equations do not possess critical set restrictions [36,43], and the damping is not necessary. The decay of solutions in such cases was also proved in [15,16] by different methods. ...
... Local stability: A perturbation argument. In this subsection, before presenting the main result of this section, we will study the asymptotic stability of the linear system associated with (43), namely, ...
Studied here is the Kawahara equation, a fifth-order Korteweg-de Vries type equation, with time-delayed internal feedback. Under suitable assumptions on the time delay coefficients, we prove that the solutions of this system are exponentially stable. First, considering a damping and delayed system, with some restriction of the spatial length of the domain, we prove that the energy of the Kawahara system goes to 0 exponentially as t → ∞. After that, by introducing a more general delayed system, and by introducing suitable energies, we show using the Lyapunov approach, that the energy of the Kawahara equation goes to zero exponentially, considering the initial data small and a restriction in the spatial length of the domain. To remove these hypotheses, we use the compactness-uniqueness argument which reduces our problem to prove an observability inequality, showing a semi-global stabilization result.
... Dispersive problems have been object of intensive research, notably in exact controllability and the stabilization of the system. About the Kawahara system in bounded domain, we can consider, for instance, the following papers: [13,15,16] and references therein. In the case with periodic boundary conditions, see [14]. ...
... The natural idea to answer the above questions is to follow closely previous articles on Kawahara system as [15,16]. However, the present system is defined on the half-line and therefore others difficulties arise as the regularity of solution and the lack of compactness. ...
... In [34,36] the damping like in (1.4) was used for (1.1) without the drift term u x . If, however, the linear term u x is dropped, both the KdV and Kawahara equations do not possess critical set restrictions [27,35], and the damping is not necessary. The decay of solutions in such case was also proved in [11,12] by different methods. ...
... Observe that the countable critical set N ⊂ R + is due to a presence of the linear drift term u x in (2.2) 1 . It is shown in [35] that N = ∅ for the model with u x neglected. We note also that a complete description of critical lengths for (2.2) is an open problem. ...
Studied here is the eventual dissipation of solutions to initial–boundary value problems for the modified Kawahara equation with and without a localized damping term included. It is shown that solutions of undamped problems posed on a bounded interval may not decay if the length of the interval is critical. In contrast, the energy associated to the locally damped problems is shown to be exponentially decreased independently of the interval length.
... Precisely, stabilization and control problems have been studied in recent years. A pioneer work is due to Silva and Vasconcellos [19]. The authors studied the stabilization of global solutions of the linear Kawahara equation in a bounded interval under the effect of a localized damping mechanism. ...
This work is devoted to present Massera-type theorems for the Kawahara system, a higher order dispersive equation, posed in a bounded domain. Precisely, thanks to some properties of the semigroup and the decays of the solutions of this equation, we are able to prove its solutions are periodic, quasi-periodic and almost periodic.
... Precisely, stabilization and control problems have been studied in recent years. A pioneer work is due to Silva and Vasconcellos [19]. The authors studied the stabilization of global solutions of the linear Kawahara equation in a bounded interval under the effect of a localized damping mechanism. ...
This work is devoted to presenting Massera-type theorems for the Kawahara system, a higher-order dispersive equation, posed in a bounded domain. Precisely, thanks to some properties of the semigroup and the decay of the solutions of this equation, we can prove its solutions are periodic, quasi-periodic, and almost periodic.
... To prove the above theorem we use multiplier techniques and the Holmgren's Uniqueness Theorem . This proof follows the same method developed in Menzala et al. [5] for the KdV linear system and Vasconcellos-Silva [11] for the Kawahara linear system. ...
This article deals with the stability problem for a higher-order dispersive model governed by the so-called Kawahara equation. To do so, a damping mechanism is introduced, which contains a distributed memory term, and then proves that the solutions of the system are exponentially stable, provided that specific assumptions on the memory kernel are fulfilled. This is possible thanks to the energy method that permits us to obtain a decay rate estimate of the energy of the problem.
We prove the exponential decay of the energy related to a locally damped fifth-order equation posed on the whole real line with the initial datum from a bounded set of L2. A local smoothing effect in H2 is established, which is essential to obtain the necessary a priory estimates. Moreover, it is shown that arguments used in the article can be applied to prove the exponential decay rate of solutions for the Korteweg-de Vries equation with a similar localized damping term provided the initial data are uniformly bounded in L2. This last fact improves some previous results.
We analyze the stabilization and the exact controllability of a third order linear equation in a bounded interval. That is, we consider the following equation:
iu_t+i\gamma u_x+ \alpha u_{xx} + i\beta u_{xxx} =0,
where u=u(x,t) is a complex valued function defined in (0,L)\times(0,+\infty) and \alpha , \beta and \gamma are real constants. Using multiplier techniques, HUM method and a special uniform continuation theorem, we prove the exponential decay of the total energy and the boundary exact controllability associated with the above equation. Moreover, we characterize a set of lengths L , named \mathcal{X} , in which it is possible to find non null solutions for the above equation with constant (in time) energy and we show it depends strongly on the parameters \alpha , \beta and \gamma .
We analyze the stabilization and internal exact control for the Kuramoto-Sivashinsky equation (KS) in a bounded interval. That is, we consider the following equation: u t +αu xx +βu xxx +ηu xxxx +uu x =0· The above equation is a model for small amplitude long waves. Using multiplier techniques we prove the exponential decay of the total energy associated with the KS system and in the linear case, using HUM method, we study its internal exact controllability.
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