NEW DECAY RATES FOR A PROBLEM OF PLATE DYNAMICS WITH FRACTIONAL DAMPING

2013 ◽  
Vol 10 (03) ◽  
pp. 563-575 ◽  
Author(s):  
RUY COIMBRA CHARÃO ◽  
CLEVERSON ROBERTO DA LUZ ◽  
RYO IKEHATA
Keyword(s):  
Decay Rates ◽  
Special Method ◽  
Rotational Inertia ◽  
Fourier Space ◽  
Plate Equation ◽  
Damping Term ◽  
Fractional Damping ◽  
Decay Properties ◽  
Optimal Decay ◽  
Additional Regularity

In this paper, we obtain decay rates for the total energy associated to the linear plate equation with effects of rotational inertia and a fractional damping term depending on a number θ ∈ [0, 1]. We observe that the dissipative structure of the equation with θ = 0 is of the regularity-loss type. This decay structure still remains true in the plate equation with a power of fractional damping θ > 0, but it becomes more weak when θ increase. This means that we can have an optimal decay estimate of solutions under an additional regularity assumption on the initial data. Our results generalize previous results by Luz and Charão and some of recent results due to Sugitani and Kawashima. We use a special method in the Fourier space which we developed in a previous work for the wave equation. So, our approach shows to be very effective to study decay properties for several problems in Rn.

scholarly journals Existence and decay rates for a semilinear dissipative fractional second order evolution equation

2020 ◽  
Vol 42 ◽  
pp. e37
Author(s):  
Ruy Coimbra Charão ◽  
Jaqueline Luiza Horbach
Keyword(s):  
Evolution Equation ◽  
Linear Problem ◽  
Second Order ◽  
Decay Rates ◽  
Rotational Inertia ◽  
Fractional Powers ◽  
Damping Term ◽  
Uniqueness Of Solutions ◽  
Fractional Damping ◽  
L2 Norm

In this work we study the existence and uniqueness of solutions and decay rates to the total energy and the L2-norm of solution for a semilinear second order evolution equation with fractional damping term and under effects of a generalized rotational inertia term in the case of plate equation. This system also includes equations of Boussinesq type that model hydrodynamic problems. We show decay rates depend- ing on the fractional powers of the operators and using an asymptotic expansion of the solution to the linear problem, we prove for some cases depending on the exponents of the operators, the optimality of the decay rates.

DECAY ESTIMATES OF SOLUTIONS TO A SEMI-LINEAR DISSIPATIVE PLATE EQUATION

2010 ◽  
Vol 07 (03) ◽  
pp. 471-501 ◽  
Author(s):  
YOUSUKE SUGITANI ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Initial Data ◽  
Sobolev Spaces ◽  
Linear Problem ◽  
Dimensional Space ◽  
Weighted Sobolev Spaces ◽  
Decay Estimates ◽  
Plate Equation ◽  
Estimates Of Solutions ◽  
Optimal Decay ◽  
Additional Regularity

We study the initial value problem for a semi-linear dissipative plate equation in n-dimensional space. We observe that the dissipative structure of the linearized equation is of the regularity-loss type. This means that we have the optimal decay estimates of solutions under the additional regularity assumption on the initial data. This regularity-loss property causes the difficulty in solving the nonlinear problem. For our semi-linear problem, this difficulty can be overcome by introducing a set of time-weighted Sobolev spaces, where the time-weights and the regularity of the Sobolev spaces are determined by our regularity-loss property. Consequently, under smallness condition on the initial data, we prove the global existence and optimal decay of the solution in the corresponding Sobolev spaces.

scholarly journals Uniform decay rates of a Bresse thermoelastic system in the whole space

2021 ◽  
Author(s):  
Mounir Afilal ◽  
Baowei Feng ◽  
Abdelaziz Soufyane
Keyword(s):  
Energy Method ◽  
Decay Rates ◽  
Lyapunov Functionals ◽  
Fourier Space ◽  
Uniform Decay ◽  
Decay Properties ◽  
Bresse System ◽  
Whole Space

In this paper, we investigate the decay properties of the thermoelastic Bresse system in the whole space. We consider many cases depending on the parameters of the model and we establish new decay rates. We need to mention here that, in some cases we don’t have the regularity-loss phenomena as in the previous works in the literature. To prove our results, we use the energy method in the Fourier space to build a very delicate Lyapunov functionals that give the desired results.

scholarly journals OPTIMAL DECAY RATES OF A NONLINEAR SUSPENSION BRIDGE WITH MEMORIES

2021 ◽  
Author(s):  
Mounir Afilal ◽  
Baowei Feng ◽  
Abdelaziz Soufyane
Keyword(s):  
Energy Method ◽  
Suspension Bridge ◽  
Decay Rates ◽  
One Dimension ◽  
Lyapunov Functionals ◽  
Decay Properties ◽  
Optimal Decay Rates ◽  
Optimal Decay

In this paper, we investigate the decay properties of suspension bridge with memories in one dimension. To prove our results, we use the energy method to build some very delicate Lyapunov functionals that give the desired results.

scholarly journals A dissipative logarithmic-Laplacian type of plate equation: Asymptotic profile and decay rates

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Ruy Coimbra Charão ◽  
Alessandra Piske ◽  
Ryo Ikehata
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Decay Rates ◽  
Plate Equation ◽  
New Model ◽  
Asymptotic Profile ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
The Cauchy Problem ◽  
Logarithmic Type

<p style='text-indent:20px;'>We introduce a new model of the logarithmic type of wave like plate equation with a nonlocal logarithmic damping mechanism. We consider the Cauchy problem for this new model in <inline-formula><tex-math id="M1">\begin{document}$ {{\bf R}}^{n} $\end{document}</tex-math></inline-formula>, and study the asymptotic profile and optimal decay rates of solutions as <inline-formula><tex-math id="M2">\begin{document}$ t \to \infty $\end{document}</tex-math></inline-formula> in <inline-formula><tex-math id="M3">\begin{document}$ L^{2} $\end{document}</tex-math></inline-formula>-sense. The operator <inline-formula><tex-math id="M4">\begin{document}$ L $\end{document}</tex-math></inline-formula> considered in this paper was first introduced to dissipate the solutions of the wave equation in the paper studied by Charão-Ikehata [<xref ref-type="bibr" rid="b7">7</xref>]. We will discuss the asymptotic property of the solution as time goes to infinity to our Cauchy problem, and in particular, we classify the property of the solutions into three parts from the viewpoint of regularity of the initial data, that is, diffusion-like, wave-like, and both of them.</p>

ASYMPTOTIC PROPERTIES FOR A SEMILINEAR PLATE EQUATION IN UNBOUNDED DOMAINS

2009 ◽  
Vol 06 (02) ◽  
pp. 269-294 ◽  
Author(s):  
CLEVERSON ROBERTO DA LUZ ◽  
RUY COIMBRA CHARÃO
Keyword(s):  
Total Energy ◽  
Decay Rate ◽  
Asymptotic Properties ◽  
Global Solutions ◽  
Decay Rates ◽  
Polynomial Decay ◽  
Rotational Inertia ◽  
Small Data ◽  
Plate Equation ◽  
Inertia Effects

We study the existence, uniqueness, and asymptotic properties of global solutions to the initial value problem associated with a semilinear, dissipative, plate equation under rotational inertia effects in ℝn. We obtain polynomial decay rate in time for the total energy. In dimension n ≥ 5 for the linear problem and n = 5 for the semilinear problem with small data, we obtain fast decay of the total energy and a decay rate t-1/2 for L2-norm of the solution and similar decay rates for the L2-norm of higher-order derivatives.

Optimal Decay Rates for Partially Dissipative Plates with Rotational Inertia

2019 ◽  
Vol 166 (1) ◽  
pp. 131-146 ◽  
Author(s):  
Fredy Maglorio Sobrado Suárez ◽  
Higidio Portillo Oquendo
Keyword(s):  
Decay Rates ◽  
Rotational Inertia ◽  
Optimal Decay Rates ◽  
Optimal Decay

scholarly journals A new stability result for a thermoelastic Bresse system with viscoelastic damping

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Soh Edwin Mukiawa ◽  
Cyril Dennis Enyi ◽  
Tijani Abdulaziz Apalara
Keyword(s):  
Stability Result ◽  
Bending Moment ◽  
Decay Rates ◽  
Viscoelastic Damping ◽  
Physical Parameters ◽  
Bresse System ◽  
Solution Energy ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Weaker Conditions

AbstractWe investigate a thermoelastic Bresse system with viscoelastic damping acting on the shear force and heat conduction acting on the bending moment. We show that with weaker conditions on the relaxation function and physical parameters, the solution energy has general and optimal decay rates. Some examples are given to illustrate the findings.

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