scholarly journals Strong Global Attractors for 3D Wave Equations with Weakly Damping

2012 ◽  
Vol 2012 ◽  
pp. 1-12 ◽  
Author(s):  
Fengjuan Meng
Keyword(s):  
Wave Equation ◽  
Global Attractor ◽  
Wave Equations ◽  
Global Attractors ◽  
Energy Space ◽  
Damped Wave Equation

We consider the existence of the global attractorA1for the 3D weakly damped wave equation. We prove thatA1is compact in(H2(Ω)∩H01(Ω))×H01(Ω)and attracts all bounded subsets of(H2(Ω)∩H01(Ω))×H01(Ω)with respect to the norm of(H2(Ω)∩H01(Ω))×H01(Ω). Furthermore, this attractor coincides with the global attractor in the weak energy spaceH01(Ω)×L2(Ω).

scholarly journals Global Attractor for Nonlinear Wave Equations with Critical Exponent on Unbounded Domain

2016 ◽  
Vol 1 (2) ◽  
pp. 581-602 ◽  
Author(s):  
Yuncheng You
Keyword(s):  
Wave Equation ◽  
Global Attractor ◽  
Nonlinear Wave ◽  
Unbounded Domain ◽  
Nonlinear Wave Equation ◽  
Wave Equations ◽  
Nonlinear Wave Equations ◽  
Growth Exponent ◽  
Global Dynamics ◽  
Dissipative Condition

AbstractAsymptotic and global dynamics of weak solutions for a damped nonlinear wave equation with a critical growth exponent on the unbounded domain ℝn(n ≥ 3) is investigated. The existence of a global attractor is proved under typical dissipative condition, which features the proof of asymptotic compactness of the solution semiflow in the energy space with critical nonlinear exponent by means of Vitali-type convergence theorem.

scholarly journals Global existence for equivalent nonlinear special scale invariant damped wave equations

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Sandra Lucente
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Vector Fields ◽  
Wave Equations ◽  
Small Data ◽  
Damped Wave Equation ◽  
Scale Invariant ◽  
Forcing Term ◽  
Time Existence

<p style='text-indent:20px;'>In this paper we give the notion of equivalent damped wave equations. As an application we study global in time existence for the solution of special scale invariant damped wave equation with small data. To gain such results, without radial assumption, we deal with Klainerman vector fields. In particular we can treat some potential behind the forcing term.</p>

scholarly journals Stabilization of damped waves on spheres and Zoll surfaces of revolution

2018 ◽  
Vol 24 (4) ◽  
pp. 1759-1788
Author(s):  
Hui Zhu
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Indicator Function ◽  
Geometric Control ◽  
Damped Wave Equation ◽  
Surfaces Of Revolution ◽  
Strong Stabilization ◽  
Geometric Objects ◽  
Dimension 2 ◽  
Appl Math

We study the strong stabilization of wave equations on some sphere-like manifolds, with rough damping terms which do not satisfy the geometric control condition posed by Rauch−Taylor [J. Rauch and M. Taylor, Commun. Pure Appl. Math. 28 (1975) 501–523] and Bardos−Lebeau−Rauch [C. Bardos, G. Lebeau and J. Rauch, SIAM J. Control Optimiz. 30 (1992) 1024–1065]. We begin with an unpublished result of G. Lebeau, which states that on 𝕊d, the indicator function of the upper hemisphere strongly stabilizes the damped wave equation, even though the equators, which are geodesics contained in the boundary of the upper hemisphere, do not enter the damping region. Then we extend this result on dimension 2, to Zoll surfaces of revolution, whose geometry is similar to that of 𝕊2. In particular, geometric objects such as the equator, and the hemi-surfaces are well defined. Our result states that the indicator function of the upper hemi-surface strongly stabilizes the damped wave equation, even though the equator, as a geodesic, does not enter the upper hemi-surface either.

Equi-exponential attraction and rate of convergence of attractors with application to a perturbed damped wave equation

2014 ◽  
Vol 144 (1) ◽  
pp. 13-51 ◽  
Author(s):  
Alexandre N. Carvalho ◽  
Jan W. Cholewa ◽  
Tomasz Dłotko
Keyword(s):  
Wave Equation ◽  
Rate Of Convergence ◽  
Global Attractors ◽  
Damped Wave Equation ◽  
Continuity Properties ◽  
Compact Semigroups ◽  
The Family ◽  
Bounded Dissipative ◽  
Attraction Property ◽  
Abstract Framework

We consider a family of bounded dissipative asymptotically compact semigroups depending on a parameter, and study the continuity properties of the corresponding family of its global attractors. We exploit the idea of the uniform exponential attraction property to discuss the continuity properties of the family of attractors and estimate the rate of convergence of the approximating attractors to the limit one. Showing a range of applications of an abstract framework, we focus much of our attention on a perturbed damped wave equation. In this latter case our results involve nonlinearities with critical exponents, for which the continuity of the family of attractors is concluded, including the rate of convergence and the regularity of the limit attractor. This complements the results in the literature.

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