Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

Fu, Xiaoyu

doi:10.1142/9789814322898_0013

Cited by 10 publications

(14 citation statements)

References 16 publications

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“…The optimal result without any geometrical hypothesis is given in . We also recall the result by Fu , where the author proved a result similar to the one in for less regular conditions (

\partial Ω \in C^{2}

) by adopting the global Carleman estimate.…”

Section: Introductionmentioning

confidence: 68%

Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

Benaissa

Rafa

2019

Mathematische Nachrichten

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“…The optimal result without any geometrical hypothesis is given in . We also recall the result by Fu , where the author proved a result similar to the one in for less regular conditions (

\partial Ω \in C^{2}

) by adopting the global Carleman estimate.…”

Section: Introductionmentioning

confidence: 68%

Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

Benaissa

Rafa

2019

Mathematische Nachrichten

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“…Observe that the semi-group property does not imply anymore that the rate of convergence in (1.2) is exponential, and there are many practical examples where the decay is only logarithmic or polynomial (see e.g. [15,16,17,28,6,19,26,32,9,12]). …”

Section: Introductionmentioning

confidence: 99%

Non-uniform stability for bounded semi-groups on Banach spaces

2008

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Let S(t) be a bounded strongly continuous semi-group on a Banach space B and −A be its generator. We say that S(t) is semi-uniformly stable when S(t)(A + 1) −1 tends to 0 in operator norm. This notion of asymptotic stability is stronger than pointwise stability, but strictly weaker than uniform stability, and generalizes the known logarithmic, polynomial and exponential stabilities.In this note we show that if S is semi-uniformly stable then the spectrum of A does not intersect the imaginary axis. The converse is already known, but we give an estimate on the rate of decay of S(t)(A + 1) −1 , linking the decay to the behaviour of the resolvent of A on the imaginary axis. This generalizes results of Lebeau and Burq (in the case of logarithmic stability) and Liu-Rao and Bátkai-Engel-Prüss-Schnaubelt (in the case of polynomial stability). BackgroundConsider a strongly continuous semi-group S(t) on a Banach space B, with generator −A. Assume that S(t) = e −tA is bounded, i.e. sup t≥0 e −tA = C < ∞.(1.1) (Throughout the article, a semi-group will be strongly continuous on [0, ∞), i.e., a C 0 -semigroup. Moreover, · will denote both the norm on B and the operator norm from B to B.) The operator A is closed and densely defined, and we denote by D(A) its domain, σ(A) its spectrum and ρ(A) its resolvent set. It is a well-known property that if (1.1) holds, then the left open half-plane {Re z < 0} is included in ρ(A) (see [25,11]). In 1988, Lyubich and Vũ [20] and Arendt and Batty [1] have shown that if σ(A) ∩ iR is countable and σ(A * ) ∩ iR contains no eigenvalue (here A * is the adjoint of A), then the semi-group is (pointwise) strongly stable, that ist→+∞ e −tA u 0 = 0. For surveys of this and other results concerning strong stability, see [4], [8] 1 .This work was partially supported by the French ANR ControlFlux. The second author would like to thank Nicolas Burq for fruitful discussions on the subject, and Luc Miller for pointing out the stability theorem of Lyubich, Vũ, Arendt and Batty and the article [3]. 1 We will only address strong stability concepts, and refer to [10] for a recent survey on different types of weak stability.

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“…Very interesting logarithmic decay results were given in [2,18] for the hyperbolic system with boundary damping under the regularity assumption that a jk (·), a(·), and ∂Ω are C ∞ -smooth. Recently, in [11], the author generalize the results of Lebeau-Robbiano [18] to the case of C 2 coefficients and a domain with C 2 boundary. However, to the best of the author's knowledge, there is no result about system (1.4) under sharp regularity assumption on the coefficients a jk (·) ∈ C 1 (Ω; R).…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

“…Here, our global Carleman estimate only needs C 1 -regularity for the coefficient a jk (·) instead of C 2 -regularity appeared in [11], which is the main novelty of this paper. Our approach, stimulated by [17] (see also [9,12,28]), is different from that in [18], which instead employed the classical local Carleman estimate and therefore needs the C ∞ -regularity of the related coefficients.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Longtime behavior of the hyperbolic equations with an arbitrary internal damping

2010

Z. Angew. Math. Phys.

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This paper is devoted to a study of the longtime behavior of the hyperbolic equations with an arbitrary internal damping, under sharp regularity assumptions that both the principal part coefficients and the boundary of the space domain (in which the system evolves) are continuously differentiable. For this purpose, we derive a new point-wise inequality for second differential operators with symmetric coefficients. Then, based on a global Carleman estimate, we establish an estimate on the underlying resolvent operator of the equation, via which, we show the logarithmic decay rate for solutions of the hyperbolic equations. (2000). Primary 93B05; Secondary 93B07 · 35B37. Mathematics Subject Classification

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Logarithmic Decay of Hyperbolic Equations with Arbitrary Small Boundary Damping

Cited by 10 publications

References 16 publications

Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

Well‐posedness and energy decay of solutions to a wave equation with a general boundary control of diffusive type

Non-uniform stability for bounded semi-groups on Banach spaces

Longtime behavior of the hyperbolic equations with an arbitrary internal damping

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