When is the energy of the 1D damped Klein-Gordon equation decaying?

Malhi, Satbir; Stanislavova, Milena

doi:10.1007/s00208-018-1725-5

Cited by 7 publications

(4 citation statements)

References 13 publications

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“…We remark that to prove the neccessity of the condition (4), the decay rates from Theorem 1 can be replaced by an a priori weaker condition, namely that there exists λ ≥ 1 such that iλ ∈ ρ(A γ ) and A γ −iλ has closed range. Then, setting µ = λ 2/s − 1, we obtain (10) for supp û ⊂ [µ − D, µ + D] (D small enough). The proof is completed analogously by taking f (x) = e iµx sin(Dx) Dx .…”

Section: Neccessity Of (4) and Threshold Valuementioning

confidence: 99%

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On the energy decay rate of the fractional wave equation on ℝ with relatively dense damping

Green

2020

Proc. Amer. Math. Soc.

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We establish upper bounds for the decay rate of the energy of the damped fractional wave equation when the averages of the damping coefficient on all intervals of a fixed length are bounded below. If the power of the fractional Laplacian, s, is between 0 and 2, the decay is polynomial. For s ≥ 2, the decay is exponential. Our assumption is also necessary for energy decay. Second, we prove that exponential decay cannot hold for s < 2 if the damping vanishes at all.Consider the following damped fractional wave equation on R for s > 0 and γ : R → R ≥0 :The damping force is represented by γw t . Herein, we study the decay rate of the energy of w, defined by

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Section: Neccessity Of (4) and Threshold Valuementioning

confidence: 99%

“…For the one-dimensional problem, this smoothness condition was relaxed by Malhi and Stanislavova in [10] to γ which are continuous and bounded. Moreover, through intricate spectral analysis, it is shown that condition (2) is equivalent to exponential decay of the energy of solutions to (1).…”

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confidence: 99%

On the energy decay rate of the fractional wave equation on ℝ with relatively dense damping

Green

2020

Proc. Amer. Math. Soc.

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“…Burq and Joly [2] proved that if γ is uniformly continuous and {γ ≥ ε} satisfies (GCC) for some ε > 0, then we have the exponential energy decay in the nonfractional case s = 2. After that, Malhi and Stanislavova [6] pointed out that (GCC) is also necessary for the exponential decay in the one-dimensional case d = 1:…”

Section: Introductionmentioning

confidence: 99%

Equivalence between the energy decay of fractional damped Klein–Gordon equations and geometric conditions for damping coefficients

Inami,

Suzuki

2023

Proc. Amer. Math. Soc. Ser. B

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We consider damped s s -fractional Klein–Gordon equations on R d \mathbb {R}^d , where s s denotes the order of the fractional Laplacian. In the one-dimensional case d = 1 d = 1 , Green (2020) established that the exponential decay for s ≥ 2 s \geq 2 and the polynomial decay of order s / ( 4 − 2 s ) s/(4-2s) hold if and only if the damping coefficient function satisfies the so-called geometric control condition. In this note, we show that the o ( 1 ) o(1) energy decay is also equivalent to these conditions in the case d = 1 d=1 . Furthermore, we extend this result to the higher-dimensional case: the logarithmic decay, the o ( 1 ) o(1) decay, and the thickness of the damping coefficient are equivalent for s ≥ 2 s \geq 2 . In addition, we also prove that the exponential decay holds for 0 > s > 2 0 > s > 2 if and only if the damping coefficient function has a positive lower bound, so in particular, we cannot expect the exponential decay under the geometric control condition.

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“…We note two recent works which have, in one dimension, utilized Fourier analysis to prove exponential decay for rough damping [9,21]. Fourier analytic methods have also proved useful in understanding (polynomial, or logarithmic) decay rates of the semi-group under weaker conditions than the GCC [1,29].…”

Section: Introductionmentioning

confidence: 99%

Uncertainty Principles Associated to Sets Satisfying the Geometric Control Condition

Green¹,

Jaye²,

Mitkovski³

2019

Preprint

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In this paper, we study forms of the uncertainty principle suggested by problems in control theory. First, we prove an analogue of the Paneah-Logvinenko-Sereda Theorem characterizing sets which satisfy the Geometric Control Condition (GCC). This result is applied to get a uniqueness result for functions with spectrum contained in sufficiently flat sets. One corollary is that a function with spectrum in an annulus of a given thickness can be bounded, in L 2 -norm, from above by its restriction to a neighborhood of a GCC set, with constant independent of the radius of the annulus. This result is applied to the energy decay rates for damped fractional wave equations.

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When is the energy of the 1D damped Klein-Gordon equation decaying?

Cited by 7 publications

References 13 publications

On the energy decay rate of the fractional wave equation on ℝ with relatively dense damping

On the energy decay rate of the fractional wave equation on ℝ with relatively dense damping

Equivalence between the energy decay of fractional damped Klein–Gordon equations and geometric conditions for damping coefficients

Uncertainty Principles Associated to Sets Satisfying the Geometric Control Condition

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