Longtime Behaviour of Strongly Damped Wave Equations, Global Attractors and Their Dimension

1991 ◽  
Vol 22 (4) ◽  
pp. 879-895 ◽  
Author(s):  
J. M. Ghidaglia ◽  
A. Marzocchi
Keyword(s):  
Wave Equations ◽  
Global Attractors ◽  
Longtime Behaviour ◽  
Strongly Damped

scholarly journals The global attractors and dimensions estimation for the Kirchho type wave equations with nonlinear strongly damped terms

2016 ◽  
Vol 12 (3) ◽  
pp. 6087-6102
Author(s):  
Chengfei Ai ◽  
Huixian Zhu ◽  
Guoguang Lin
Keyword(s):  
Global Attractor ◽  
Wave Equations ◽  
Global Attractors ◽  
Time Behavior ◽  
Priori Estimate ◽  
Type Wave ◽  
Long Time ◽  
Uniqueness Of The Solution ◽  
The Galerkin Method ◽  
Strongly Damped

This paper studies the long time behavior of the solution to the initial boundaryvalue problems for a class of strongly damped Kirchho type wave equations:utt "1ut + j ut jp1 ut + j u jq1 u (kruk2)u = f(x):Firstly, we prove the existence and uniqueness of the solution by priori estimate and the Galerkin method. Then we obtain to the existence of the global attractor. Finally, we consider that the estimation of the upper bounds of Hausdor and fractal dimensionsfor the global attractor is obtained.

scholarly journals Quasi-stability and continuity of attractors for nonlinear system of wave equations

2021 ◽  
Vol 8 (1) ◽  
pp. 27-45
Author(s):  
M. M. Freitas ◽  
M. J. Dos Santos ◽  
A. J. A. Ramos ◽  
M. S. Vinhote ◽  
M. L. Santos
Keyword(s):  
Wave Equations ◽  
Coupled System ◽  
Global Attractors ◽  
Time Behavior ◽  
Exponential Attractors ◽  
Small Perturbations ◽  
Recent Approach ◽  
Nonlinear Coupled System ◽  
Long Time ◽  
External Forces

Abstract In this paper, we study the long-time behavior of a nonlinear coupled system of wave equations with damping terms and subjected to small perturbations of autonomous external forces. Using the recent approach by Chueshov and Lasiecka in [21], we prove that this dynamical system is quasi-stable by establishing a quasistability estimate, as consequence, the existence of global and exponential attractors is proved. Finally, we investigate the upper and lower semicontinuity of global attractors under autonomous perturbations.

scholarly journals On the existence of solutions of strongly damped nonlinear wave equations

2000 ◽  
Vol 23 (6) ◽  
pp. 369-382 ◽  
Author(s):  
Jong Yeoul Park ◽  
Jeong Ja Bae
Keyword(s):  
Nonlinear Wave ◽  
Existence And Uniqueness ◽  
Wave Equations ◽  
Existence Of Solutions ◽  
Hyperbolic Type ◽  
Nonlinear Wave Equations ◽  
Gamma Delta ◽  
Uniqueness Of Solutions ◽  
Alpha Beta ◽  
Strongly Damped

We investigate the existence and uniqueness of solutions of the following equation of hyperbolic type with a strong dissipation:utt(t,x)−(α+β(∫Ω|∇u(t,y)|2dy)γ)Δu(t,x)                                −λΔut(t,x)+μ|u(t,x)|q−1u(t,x)=0,     x∈Ω,t≥0            u(0,x)=u0(x),          ut(0,x)=u1(x),      x∈Ω,  u|∂Ω=0, whereq>1,λ>0,μ∈ℝ,α,β≥0,α+β>0, andΔis the Laplacian inℝN.

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