Asymptotic Stability and Energy Decay Rates for Solutions of the Wave Equation with Memory in a Star-Shaped Domain

2000 ◽  
Vol 38 (5) ◽  
pp. 1581-1602 ◽  
Author(s):  
M. Aassila ◽  
M. M. Cavalcanti ◽  
J. A. Soriano
Keyword(s):  
Wave Equation ◽  
Asymptotic Stability ◽  
Energy Decay ◽  
Decay Rates ◽  
Energy Decay Rates ◽  
Shaped Domain ◽  
With Memory

Asymptotic stability and energy decay rates for solutions of hyperbolic equations with boundary damping

1977 ◽  
Vol 77 (1-2) ◽  
pp. 97-127 ◽  
Author(s):  
John P. Quinn ◽  
David L. Russell
Keyword(s):  
Wave Equation ◽  
Asymptotic Stability ◽  
Asymptotic Behaviour ◽  
Hyperbolic Equations ◽  
Energy Decay ◽  
Decay Rates ◽  
The Other ◽  
Boundary Damping ◽  
Energy Absorbing ◽  
Energy Decay Rates

SynopsisThis report deals with the asymptotic behaviour of solutions of the wave equation in a domain Ω ⊆Rn. The boundary, Γof Ωft consists of two parts. One part reflects all energy while the other part absorbs energy to a degree. If the energy-absorbing part is non-empty we show that the energy tends to zero as t→∞. With stronger assumptions we are able to obtain decay rates for the energy. Certain relationships with controlability are discussed and used to advantage.

Sharp decay rates for a class of nonlinear viscoelastic plate models

2017 ◽  
Vol 20 (02) ◽  
pp. 1750010 ◽  
Author(s):  
E. H. Gomes Tavares ◽  
M. A. Jorge Silva ◽  
T. F. Ma
Keyword(s):  
Decay Rate ◽  
Evolution Equations ◽  
Viscoelastic Plate ◽  
Energy Decay ◽  
Decay Rates ◽  
Memory Kernel ◽  
Energy Decay Rate ◽  
Plate Equations ◽  
Energy Decay Rates ◽  
With Memory

This paper is concerned with uniform stability of the energy corresponding to a class of nonlinear plate equations with memory. It is assumed that the memory kernel [Formula: see text] satisfies the condition [Formula: see text] of Alabau-Boussouira and Cannarsa [A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I 347 (2009) 867–872], where [Formula: see text] is positive, convex, increasing, and satisfies [Formula: see text]. Then, we obtain sharp energy decay rate in the sense that it recovers the decay rate assumed to the memory kernel. To this end we use a recent approach proposed by Lasiecka and Wang [Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer Series INDAM, Vol. 10 (Springer, 2014), pp. 271–303].

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