Uniform decay rate estimates for the semilinear wave equation in inhomogeneous medium with locally distributed nonlinear damping

Cavalcanti, Marcelo M.; Cavalcanti, V. N. Domingos; Fukuoka, Ryuichi; Pampu, Ademir; Astudillo, María

doi:10.1088/1361-6544/aac75d

Cited by 26 publications

(9 citation statements)

References 39 publications

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“…The equation (1.6) has been considered in several papers, see for instance [19,20,23,29,14,21,1,13], the review paper [30] and the recent monograph [22]. For the homogeneous boundary conditions (u = 0 at both ends, corresponding to J b = 0), it is well known that the initial-boundary value problem for (1.6) is well-posed for initial data (u 0 , ∂ t u 0 ) ∈ H 1 0 (I) × L 2 (I), for k(x) ∈ L ∞ (I) with k(x) ≥ 0, and decay estimates for the energy are obtained, either exponential or polynomial.…”

Section: Introductionmentioning

confidence: 99%

Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain

Amadori

Aqel

Santo

2019

Journal de Mathématiques Pures et Appliquées

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In this paper we study the long time behavior for a semilinear wave equation with spacedependent and nonlinear damping term. After rewriting the equation as a first order system, we define a class of approximate solutions that employ tipical tools of hyperbolic systems of conservation laws, such as the Riemann problem. By recasting the problem as a discretetime nonhomogeneous system, which is related to a probabilistic interpretation of the solution, we provide a strategy to study its long-time behavior uniformly with respect to the mesh size parameter ∆x = 1/N → 0. The proof makes use of the Birkhoff decomposition of doubly stochastic matrices and of accurate estimates on the iteration system as N → ∞.Under appropriate assumptions on the nonlinearity, we prove the exponential convergence in L ∞ of the solution to the first order system towards a stationary solution, as t → +∞, as well as uniform error estimates for the approximate solutions.

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Section: Introductionmentioning

confidence: 99%

Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain

Amadori

Aqel

Santo

2019

Journal de Mathématiques Pures et Appliquées

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“…The case of the semilinear wave equation as in the present article also attracted the attention of many researchers. However, most results assume that the damping is either linear (see e.g., [20], [21], [29], [31], [38], and [45]) or linearly bounded [10]. In the present article, we generalize the known results mentioned on the subject to a considerably larger class of dissipative effects that are not necessarily linear or linearly bounded.…”

Section: Introductionmentioning

confidence: 58%

“…An advantage of the present approach is that we do not need to consider the well known linearizability property due to Gérard [24], which, roughly speaking, guarantees that the semilinear (subcritical) solution is asymptotically close to the solution of the undamped linear wave equation. This work uses the above approach to deal with a larger class of dissipative effects, thus substantially generalizing the recent results of [10].…”

Section: Introductionmentioning

confidence: 92%

“…The uniqueness of solution for the range 1 ≤ p < n+2 n−2 follows the same idea in [10] and consequently will be omitted.…”

Section: )mentioning

confidence: 99%

“…holds true for all T ≥ T 0 (for a certain T 0 associated with the geometric control condition). The approach based on the above ingredients is a well known strategy in stabilization theory; see for example [10], [20], [21], [29], [45] and references therein. These papers work directly on the original problem (1.1) and use a unique continuation argument as a crucial ingredient for proving the observability inequality.…”

Section: Introductionmentioning

confidence: 99%

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Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation

Cavalcanti¹,

Cavalcanti²,

Martinez³

et al. 2020

Preprint

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We study the stabilization and the wellposedness of solutions of the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation. The novelty of this paper is that we deal with the difficulty that the main equation does not have good nonlinear structure amenable to a direct proof of a priori bounds and a desirable observability inequality. It is well known that observability inequalities play a critical role in characterizing the long time behaviour of solutions of evolution equations, which is the main goal of this study. In order to address this, we truncate the nonlinearities, and thereby construct approximate solutions for which it is possible to obtain a priori bounds and prove the essential observability inequality. The treatment of these approximate solutions is still a challenging task and requires the use of Strichartz estimates and some microlocal analysis tools such as microlocal defect measures. We include an appendix on the latter topic here to make the article self contained and supplement details to proofs of some of the theorems which can be found in the lecture notes of [7]. Once we establish essential observability properties for the approximate solutions, it is not difficult to prove that the solution of the original problem also possesses a similar feature via a delicate passage to limit. In the last part of the paper, we establish various decay rate estimates for different growth conditions on the nonlinear dissipative effect. We in particular generalize the known results on the subject to a considerably larger class of dissipative effects.

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Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

Carrião

Miyagaki

Vicente

2022

Mathematische Nachrichten

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In this paper, we study the exponential decay of the energy associated to an initial value problem involving the wave equation on the hyperbolic space 𝔹 𝑁 . The space 𝔹 𝑁 is the unit disc {𝑥 ∈ ℝ 𝑁 ∶ |𝑥| < 1} of ℝ 𝑁 endowed with the Riemannian metric 𝑔 given by 𝑔 𝑖𝑗 = 𝑝 2 𝛿 𝑖𝑗 , where 𝑝(𝑥) = 2 1−|𝑥| 2 and 𝛿 𝑖𝑗 = 1, if 𝑖 = 𝑗 and 𝛿 𝑖𝑗 = 0, if 𝑖 ≠ 𝑗. Making an appropriate change, the problem can be seen as a singular problem on the boundary of the open ball 𝐵 1 = {𝑥 ∈ ℝ 𝑁 ; |𝑥| < 1} endowed with the euclidean metric. The proof is based on the multiplier techniques combined with the use of Hardy's inequality, in a version due to the Brezis-Marcus, which allows us to overcome the difficulty involving the singularities.

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Uniform decay rate estimates for the semilinear wave equation in inhomogeneous medium with locally distributed nonlinear damping

Cited by 26 publications

References 39 publications

Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain

Decay of approximate solutions for the damped semilinear wave equation on a bounded 1d domain

Decay rate estimates for the wave equation with subcritical semilinearities and locally distributed nonlinear dissipation

Exponential decay for semilinear wave equation with localized damping in the hyperbolic space

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