Energy decay for the linear Klein–Gordon equation and boundary control

Nunes, Ruikson S. O.; Bastos, Waldemar D.

doi:10.1016/j.jmaa.2014.01.053

Cited by 17 publications

(25 citation statements)

References 9 publications

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“…If one considers the following weakly damped Klein‐Gordon equation in place of ,

u_{t t} (t, x) - Δ u (t, x) + m^{2} u (t, x) + u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times {boldR}^{n},

one has many previous papers, and one can cite D'Abbicco, da‐Luz‐Ikehata‐Charão, Zuazua, and the references therein (for nondamped Klein‐Gordon equations, one can cite several papers due to Nunes‐Bastos and the references therein concerning the local energy decay property and its related research). In this connection, in D'Abbicco, a more generalized evolution equations are studied:

u_{t t} false(t, x false) + {false(- normalΔ false)}^{σ} false(t, x false) + 2 a {false(- normalΔ false)}^{α} u_{t} false(t, x false) + m^{2} {false(- normalΔ false)}^{μ} u false(t, x false) = 0, false(t, x false) \in false(0, \infty false) \times R^{n};

however, his result due to D'Abbicco, , theorem 4.1 which is close relation to ours, does not investigate the case of σ = α = 1 and μ = 0 much less decay estimates from below.…”

Section: Introductionmentioning

confidence: 78%

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

D’Abbicco

Ikehata

2019

Math Methods in App Sciences

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We consider the Cauchy problem in Rn for strongly damped Klein‐Gordon equations. We derive asymptotic profiles of solutions with weighted L1,1(Rn) initial data by a simple method introduced by the second author. Furthermore, from the obtained asymptotic profile, we get the optimal decay order of the L2‐norm of solutions. The obtained results show that the wave effect will be relatively weak because of the mass term, especially in the low‐dimensional case (n = 1,2) as compared with the strongly damped wave equations without mass term (m = 0), so the most interesting topic in this paper is the n = 1,2 cases to compare the difference.

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“…If one considers the following weakly damped Klein‐Gordon equation in place of ,

u_{t t} (t, x) - Δ u (t, x) + m^{2} u (t, x) + u_{t} (t, x) = 0, (t, x) \in (0, \infty) \times {boldR}^{n},

u_{t t} false(t, x false) + {false(- normalΔ false)}^{σ} false(t, x false) + 2 a {false(- normalΔ false)}^{α} u_{t} false(t, x false) + m^{2} {false(- normalΔ false)}^{μ} u false(t, x false) = 0, false(t, x false) \in false(0, \infty false) \times R^{n};

however, his result due to D'Abbicco, , theorem 4.1 which is close relation to ours, does not investigate the case of σ = α = 1 and μ = 0 much less decay estimates from below.…”

Section: Introductionmentioning

confidence: 78%

Asymptotic profile of solutions for strongly damped Klein‐Gordon equations

D’Abbicco

Ikehata

2019

Math Methods in App Sciences

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“…Furthermore, a recent result by M. Malloug [13] which establishes an exponential decay of the local energy for the damped Klein-Gordon equation in exterior domain and R-S-O. Nunes and W-D. Bastos [15] obtains polynomial decay of the local energy for the linear Klein Gordon equation.…”

Section: Introduction and Position Of The Problemmentioning

confidence: 92%

“…In section 3, we prove the exponential decay of the localized linear Klein-Gordon equation. For this purpose, we introduce the Lax-Phillips semigroup Z KG (t) and an argument inspired from the work of [15]. Section 4 is devoted to prove the main result of this paper.…”

Section: Introduction and Position Of The Problemmentioning

confidence: 99%

Decay of the local energy for the solutions of the critical Klein–Gordon equation

Bchatnia

2019

Semigroup Forum

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In this paper, we prove the exponential decay of local energy for the Klein-Gordon equation with localized critical nonlinearity. The proof relies on generalized Strichartz estimates, and semi-group of Lax-Phillips.Résumé: Dans cet article, on démontre la décroissance exponentielle de l'énergie locale des solutions de l'équation de Klein-Gordon, avec une nonlinéarité critique localisée. La preuve est basée sur les inégalités de Strichartz généralisées, et le semi-groupe de Lax-Phillips.

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“…As usual, in the study of hyperbolic equations, it is important to know how the energy behaves as t →+ ∞ . In Nunes and Bastos, local decay of energy for the solution of a single linear Klein‐Gordon equation was obtained. In the present paper, we extend it to the solution of .…”

Section: Introductionmentioning

confidence: 99%

“…In Rajaram and Najafi, the method HUM presented in Lions, was used to obtain the desirable control results. Here, as in Bastos et al and Nunes and Bastos, we use the controllability method presented by D.L. Russell .…”

Section: Introductionmentioning

confidence: 99%