Convexity and Weighted Integral Inequalities for Energy Decay Rates of Nonlinear Dissipative Hyperbolic Systems

2004 ◽  
Vol 51 (1) ◽  
pp. 61-105 ◽  
Author(s):  
Fatiha Alabau-Boussouira
Keyword(s):  
Hyperbolic Systems ◽  
Energy Decay ◽  
Decay Rates ◽  
Integral Inequalities ◽  
Weighted Integral ◽  
Energy Decay Rates

scholarly journals Uniform Energy Decay Rates for the Fuzzy Viscoelastic Model with a Nonlinear Source

2020 ◽  
Vol 2020 ◽  
pp. 1-12
Author(s):  
Fengyun Zhang
Keyword(s):  
Viscoelastic Model ◽  
Energy Decay ◽  
Decay Rates ◽  
Polynomial Form ◽  
Computational Technique ◽  
Exponential Form ◽  
Nonlinear Source ◽  
Priori Estimates ◽  
Energy Decay Rates ◽  
Mathematica Software

This paper considers the fuzzy viscoelastic model with a nonlinear source u t t + L u + ∫ 0 t g t − ζ Δ u ζ d ζ − u γ u − η Δ u t = 0 in a bounded field Ω. Under weak assumptions of the function g t , with the aid of Mathematica software, the computational technique is used to construct the auxiliary functionals and precise priori estimates. As time goes to infinity, we prove that the solution is global and energy decays to zero in two different ways: the exponential form and the polynomial form.

scholarly journals Energy decay rates of magnetoelastic waves in a bounded conductive medium

2009 ◽  
Vol 25 (3) ◽  
pp. 797-821 ◽  
Author(s):  
Ruy Coimbra Charão ◽  
◽  
Jáuber Cavalcante Oliveira ◽  
Gustavo Alberto Perla Menzala ◽  
Keyword(s):  
Energy Decay ◽  
Decay Rates ◽  
Conductive Medium ◽  
Energy Decay Rates ◽  
Magnetoelastic Waves

scholarly journals Optimal energy decay rates for abstract second order evolution equations with nonautonomous damping

2021 ◽  
Author(s):  
J. R. Luo ◽  
T. J. Xiao
Keyword(s):  
Evolution Equation ◽  
Polynomial Growth ◽  
Evolution Equations ◽  
Energy Decay ◽  
Second Order ◽  
Decay Rates ◽  
Optimal Decay ◽  
Optimal Decay Rate ◽  
Energy Decay Rates ◽  
Autonomous Evolution

We consider an abstract second order non-autonomous evolution equation in a Hilbert space $H:$ $u''+Au+\gamma(t) u'+f(u)=0,$ where $A$ is a self-adjoint and nonnegative operator on $H$, $f$ is a conservative $H$-valued function with polynomial growth (not necessarily to be monotone), and $\gamma(t)u'$ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient $\gamma(t)$ and the exponent associated with the nonlinear term $f$? There seems to be little development on the study of such problems, with regard to {\it non-autonomous} equations, even for strongly positive operator $A$. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of $\gamma(t)$ and $f$.  As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when $f$ is a monotone operator.

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