A general method for proving sharp energy decay rates for memory-dissipative evolution equations

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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Sharp decay rates for a class of nonlinear viscoelastic plate models

Communications in Contemporary Mathematics ◽

10.1142/s0219199717500109 ◽

2017 ◽

Vol 20 (02) ◽

pp. 1750010 ◽

Cited By ~ 3

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

Journal of Mathematics ◽

10.1155/2020/8615149 ◽

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.

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