Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity

Yang, Meihua; Sun, Chunyou

doi:10.1090/s0002-9947-08-04680-1

Cited by 47 publications

(36 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…For the nonlinear function f satisfying (2.1)-(2.2), from [1] (see also [13,23,28,35] for our situation) we know that f allows the following decomposition f = f 0 + f 1 , where f 0 , f 1 ∈ C 1 (R) and satisfy…”

Section: Decomposition Of the Equationmentioning

confidence: 99%

“…We will follow the idea (method) in [23,28,35,37] to deduce the asymptotic regularity. Decomposing the solution S ε (t)(u 0 , v 0 ) = (u(t), u t (t)) into the sum…”

Section: Decomposition Of the Equationmentioning

confidence: 99%

“…where both k 1 and Q(·) are independent of ε ∈ [0, 1], and σ = Based on Lemma 5.1 and Lemma 5.3, following the idea in Zelik [37], we can now decompose u(t) as follows (the proof is completely similar to that in [28,35,37] since the estimates in Lemmas 5.1 and 5.3 hold uniformly w.r.t. ε ∈ [0, 1]):…”

Section: Lemma 52 Let {S(t)} T≥0 Be a Continuous Semigroup On The Bmentioning

confidence: 99%

“…for all t ≥ 0. , there are at least two ways to prove this lemma: one is as that in [28,35] to apply the idea introduced in Zelik [37]; another one is the method introduced recently in Conti and Pata [13]. Here we will use the method in [13].…”

Section: Lemma 52 Let {S(t)} T≥0 Be a Continuous Semigroup On The Bmentioning

confidence: 99%

“…We study the long-time behavior of the following semilinear evolution equation of second order in time: When ε = 0, (E 0 ) is the usual strongly damped wave equation, and its asymptotic behavior has been studied extensively in terms of attractors; see [4,5,7,13,16,23,25,32,35,36].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Asymptotic behavior for a semilinear second order evolution equation

Sun

Yang

Duan

2011

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. This paper is devoted to the qualitative analysis for a second order evolution equation u tt −Δu−Δu t −εΔu tt +f (u) = g(x) (ε ∈ [0, 1]) with critical nonlinearity. Some uniformly (w.r.t. ε ∈ [0, 1]) asymptotic regularity about the solutions has been established for both g(x) ∈ L 2 (Ω) and g(x) ∈ H −1 , which shows that the solutions are exponentially approaching a more regular fixed subset uniformly (w.r.t. ε ∈ [0, 1]). As an application of this regularity result, a family {E ε } ε∈ [0,1] of finite dimensional exponential attractors has been constructed. Moreover, to characterize the relation with a strongly damped wave equation (ε = 0), the upper semicontinuity, at ε = 0, of the global attractors has been proved.

show abstract

Section: Decomposition Of the Equationmentioning

confidence: 99%

“…We will follow the idea (method) in [23,28,35,37] to deduce the asymptotic regularity. Decomposing the solution S ε (t)(u 0 , v 0 ) = (u(t), u t (t)) into the sum…”

Section: Decomposition Of the Equationmentioning

confidence: 99%

Section: Lemma 52 Let {S(t)} T≥0 Be a Continuous Semigroup On The Bmentioning

confidence: 99%

Section: Lemma 52 Let {S(t)} T≥0 Be a Continuous Semigroup On The Bmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic behavior for a semilinear second order evolution equation

Sun

Yang

Duan

2011

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

show abstract

Qualitative theory for strongly damped wave equations

Azevedo

Cuevas

Soto

2017

Math Methods in App Sciences

View full text Add to dashboard Cite

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→∞

show abstract

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Zhang

2023

Mediterr. J. Math.

View full text Add to dashboard Cite

Dynamics of strongly damped wave equations in locally uniform spaces: Attractors and asymptotic regularity

Cited by 47 publications

References 31 publications

Asymptotic behavior for a semilinear second order evolution equation

Asymptotic behavior for a semilinear second order evolution equation

Qualitative theory for strongly damped wave equations

Bi-space Global Attractors for a Class of Second-Order Evolution Equations with Dispersive and Dissipative Terms in Locally Uniform Spaces

Contact Info

Product

Resources

About