Asymptotic Periodicity for Strongly Damped Wave Equations

Cuevas, Claudio; Lizama, Carlos; Soto, Herme

doi:10.1155/2013/308616

Cited by 4 publications

(5 citation statements)

References 34 publications

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“…An example of mathematical model represented in the form is the wave equation with structural damping (see Carvalho and Cholewa and Chen and Triggiani). This manuscript is a natural continuation of the work of Cuevas et al, which investigates the existence and uniqueness of asymptotically almost‐periodic mild solutions for strongly damped wave equations of type . Here we are concerned with the asymptotic periodicity, L p ‐boundedness properties, classical (resp., strong) solutions, and the topological structure of solutions set of problem .…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 94%

“…The strongly damped wave equations are of a great current interest, and they have been investigated in several contexts by many mathematicians under different assumptions. The literature offers topics such as existence, long time asymptotic behavior, attractors, well‐posedness, decay estimates, blowup, controllability, bootstrapping and regularity, and asymptotic periodicity . These equations are also considered in physical areas, such as heat conduction and solid mechanics.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

“…It is relevant to know when the set of mild solutions of enjoy key topological properties. This type of information has not been analyzed in the paper . In such direction, we have the following results.…”

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

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Qualitative theory for strongly damped wave equations

Azevedo

Cuevas

Soto

2017

Math Methods in App Sciences

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We investigate the asymptotic periodicity, L p -boundedness, classical (resp., strong) solutions, and the topological structure of solutions set of strongly damped semilinear wave equations. The theoretical results are well complemented with a set of very illustrating applications. KEYWORDSasymptotic behavior, boundedness, classical solutions, damped wave equations, strong solutions, topological structure of solutions set where X 1 2 is the fractional power space associated with A as in the work of Henry. 19 Equations like (1.1) have a lot of nontrivial and interesting features and appear in the literature under the name of strongly damped wave equations. An example of mathematical model represented in the form (1.1) is the wave equation with structural damping (see Carvalho and Cholewa 9 and Chen and Triggiani [20][21][22] ). This manuscript is a natural continuation of the work of Cuevas et al, 18 which investigates the existence and uniqueness of asymptotically almost-periodic mild solutions for strongly damped wave equations of type 1.1. Here we are concerned with the asymptotic periodicity, L p -boundedness properties, classical (resp., strong) solutions, and the topologicalProof. The proof requires straightforward modifications in the proof of Lemma 4.1 by Andrede et al. 27 We omit the details.Throughout this paper we always assume that ( , A) is an admissible pair. Pseudo S-asymptotically -periodic functionsLet Y be an arbitrary Banach space. In this work C b ([0, ∞); Y) denotes the space consisting of the continuous and bounded functions from [0, ∞) into Y, endowed with the norm of the uniform convergence. Definition 2.3. (Pierri and Rolnik[23]) A function f ∈ C b ([0, ∞); Y) is called pseudo S-asymptotically periodic if there is > 0 such that lim h→∞

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Section: Introduction and Statements Of Resultsmentioning

confidence: 94%

Section: Introduction and Statements Of Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Qualitative theory for strongly damped wave equations

Azevedo

Cuevas

Soto

2017

Math Methods in App Sciences

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“…There has been much recent work regarding strongly damped wave equations. For instance, well-posedness of the problem is discussed in [7,20,22], asymptotic behavior in [3,8,31,35] solution blowup in [2,12], and decay estimates in [18]. In particular, FEM for the strongly damped wave equation has been analyzed in [25] using the Ritz-Volterra projection, and [24] uses the classical Ritz-projection in the homogeneous case with Rayleigh damping.…”

Section: Introductionmentioning

confidence: 99%

A generalized finite element method for the strongly damped wave equation with rapidly varying data

2021

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We propose a generalized finite element method for the strongly damped wave equation with highly varying coefficients. The proposed method is based on the localized orthogonal decomposition, and is designed to handle independent variations in both the damping and the wave propagation speed respectively. The method does so by automatically correcting for the damping in the transient phase and for the propagation speed in the steady state phase. Convergence of optimal order is proven, independent of the derivatives of the coefficients. We present numerical examples that confirm the theoretical findings.

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“…Cuevas et al in [9] studied asymptotic almost periodic mild solutions of problem (1.1). However, to the best of our knowledge, the fact that the existence and uniqueness of weighted pseudo almost automorphic mild solutions to (1.1) with the forcing term f belong to the space of Stepanov weighted pseudo almost automorphic functions is an untreated original problem, which constitutes one of the main motivations of this work.…”

Section: Introductionmentioning

confidence: 99%