On the uniform decay for the Korteweg–de Vries equation with weak damping

Massarolo, C. P.; Menzala, G. Perla; Pazoto, Ademir F.

doi:10.1002/mma.847

Cited by 31 publications

(17 citation statements)

References 21 publications

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“…In Massarolo et al [14] has been obtained, for KdV system, the locally uniform energy decay at an exponential rate with a very weak localized dissipation namely the H −1 norm. Indeed, they considered the damping given by the bounded linear operator B = 1 ω (− ) −1 , where ω is a non-empty open subset of (0, L).…”

Section: Final Remarksmentioning

confidence: 98%

Stabilization of the Kawahara equation with localized damping

Vasconcellos

Silva

2009

ESAIM: COCV

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Abstract.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.Mathematics Subject Classification. 35Q35, 35B40, 35Q53.

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Section: Final Remarksmentioning

confidence: 98%

Stabilization of the Kawahara equation with localized damping

Vasconcellos

Silva

2009

ESAIM: COCV

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“…where ω is a nonempty open subset of (0, L), was considered in [14]. The corresponding techniques introduced by Pazoto in [19] were properly adapted in this case which could be thought of as a localized weak dissipation (in the H −1 −norm) of the scalar KdV.…”

Section: ||U (· T)||mentioning

confidence: 99%

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

2011

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Abstract. We study the stabilization of solutions of a coupled system of Kortewegde Vries (KdV) equations in a bounded interval under the effect of a localized damping term. We use multiplier techniques combined with the so-called "compactness-uniqueness argument". The problem is then reduced to proving a unique continuation property (UCP) for weak solutions. The exponential decay of solutions was previously obtained in Bisognin, Bisognin, and Menzala (2003) when the damping was effective simultaneously in neighborhoods of both extremes of the bounded interval. In this work we address the general case using a different approach to obtain the UCP and stabilize the system.

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“…There is much literature concerned with the energy decay rate estimate for other related problems, for more recent results, we refer the reader to [4,6,9,12,15,21,25,28] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Behavior for Petrovsky Equation with Localized Damping

Han

Wang

2009

Acta Appl Math

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In this paper, we investigate asymptotic behavior for the solution of the Petrovsky equation with locally distributed damping. Without growth condition on the damping at the origin, we extend the energy decay result in Martinez (Rev. Math. Complut. Madr. 12(1):251-283, 1999) for the single wave equation to the Petrovsky equation. The explicit energy decay rate is established by using piecewise multiplier techniques and weighted nonlinear integral inequalities.

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On the uniform decay for the Korteweg–de Vries equation with weak damping

Cited by 31 publications

References 21 publications

Stabilization of the Kawahara equation with localized damping

Stabilization of the Kawahara equation with localized damping

Uniform stabilization of a class of coupled systems of KdV equations with localized damping

Asymptotic Behavior for Petrovsky Equation with Localized Damping

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