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 Uniform decay rates of a coupled suspension bridges with temperature

2021 ◽  
Author(s):  
Mounir Afilal ◽  
Mohamed Alahyane ◽  
Abdelaziz Soufyane
Keyword(s):  
Energy Method ◽  
Lyapunov Functional ◽  
Decay Rates ◽  
Suspension Bridges ◽  
Uniform Decay ◽  
Decay Properties

AbstractIn this paper, we investigate the decay properties of the thermoelastic suspension bridges model. We prove that the energy is decaying exponentially. To our knowledge, our result is new and our method of proof is based on the energy method to build the appropriate Lyapunov functional.

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 On Existence, Uniform Decay Rates, and Blow-Up for Solutions of a Nonlinear Wave Equation with Dissipative and Source

2012 ◽  
Vol 2012 ◽  
pp. 1-27
Author(s):  
Xiaopan Liu
Keyword(s):  
Wave Equation ◽  
Global Existence ◽  
Nonlinear Wave Equation ◽  
Energy Method ◽  
Blow Up ◽  
Existence Of Solution ◽  
Decay Rates ◽  
Nonlinear Hyperbolic Equation ◽  
Asymptotic Behaviors ◽  
Uniform Decay

This paper studies the blow-up and existence, and asymptotic behaviors of the solution of a nonlinear hyperbolic equation with dissipative and source terms. By using Galerkin procedure and the perturbed energy method, the local and global existence of solution is established. In addition, by the concave method, the blow-up of solutions can be obtained.

The asymptotic behavior of the Bresse–Cattaneo system

2016 ◽  
Vol 18 (04) ◽  
pp. 1550045 ◽  
Author(s):  
Belkacem Said-Houari ◽  
Taklit Hamadouche
Keyword(s):  
Heat Conduction ◽  
Differential Equations ◽  
Decay Rate ◽  
Stability Number ◽  
Timoshenko System ◽  
Decay Properties ◽  
Bresse System ◽  
Whole Space ◽  
The Stability ◽  
The One

In this paper, we investigate the decay properties of the Bresse–Cattaneo system in the whole space. We show that the coupling of the Bresse system with the heat conduction of the Cattaneo theory leads to a loss of regularity of the solution and we prove that the decay rate of the solution is very slow. In fact, we show that the [Formula: see text]-norm of the solution decays with the rate of [Formula: see text]. The behavior of solutions depends on a certain number [Formula: see text] (which is the same stability number for the Timoshenko–Cattaneo system [Damping by heat conduction in the Timoshenko system: Fourier and Cattaneo are the same, J. Differential Equations 255(4) (2013) 611–632; The stability number of the Timoshenko system with second sound, J. Differential Equations 253(9) (2012) 2715–2733]) which is a function of the parameters of the system. In addition, we show that we obtain the same decay rate as the one of the solution for the Bresse–Fourier model [The Bresse system in thermoelasticity, to appear in Math. Methods Appl. Sci.].

DECAY PROPERTY FOR THE TIMOSHENKO SYSTEM WITH FOURIER'S TYPE HEAT CONDUCTION

2014 ◽  
Vol 11 (01) ◽  
pp. 135-157 ◽  
Author(s):  
NAOFUMI MORI ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Heat Conduction ◽  
Energy Method ◽  
Decay Estimates ◽  
Fourier Space ◽  
Pointwise Estimates ◽  
Timoshenko System ◽  
One Dimensional ◽  
Estimates Of Solutions ◽  
Whole Space ◽  
The One

We study the Timoshenko system with Fourier's type heat conduction in the one-dimensional (whole) space. We observe that the dissipative structure of the system is of the regularity-loss type, which is somewhat different from that of the dissipative Timoshenko system studied earlier by Ide–Haramoto–Kawashima. Moreover, we establish optimal L2decay estimates for general solutions. The proof is based on detailed pointwise estimates of solutions in the Fourier space. Also, we introuce here a refinement of the energy method employed by Ide–Haramoto–Kawashima for the dissipative Timoshenko system, which leads us to an improvement on their energy method.

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 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.

Stabilization of the wave equation with a nonlinear delay term in the boundary conditions

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Wassila Ghecham ◽  
Salah-Eddine Rebiai ◽  
Fatima Zohra Sidiali
Keyword(s):  
Wave Equation ◽  
Original Problem ◽  
Existence Of Solutions ◽  
Semigroup Theory ◽  
Decay Rates ◽  
Uniform Decay ◽  
Delay Term ◽  
Nonlinear Semigroup Theory ◽  
Nonlinear Boundary Feedback ◽  
Nonlinear Delay

Abstract A wave equation in a bounded and smooth domain of ℝ n {\mathbb{R}^{n}} with a delay term in the nonlinear boundary feedback is considered. Under suitable assumptions, global existence and uniform decay rates for the solutions are established. The proof of existence of solutions relies on a construction of suitable approximating problems for which the existence of the unique solution will be established using nonlinear semigroup theory and then passage to the limit gives the existence of solutions to the original problem. The uniform decay rates for the solutions are obtained by proving certain integral inequalities for the energy function and by establishing a comparison theorem which relates the asymptotic behavior of the energy and of the solutions to an appropriate dissipative ordinary differential equation.

scholarly journals Large Time Decay of Solutions to a Linear Nonautonomous System in Exterior Domains

Mathematics ◽  
2021 ◽  
Vol 9 (8) ◽  
pp. 841
Author(s):  
Toshiaki Hishida
Keyword(s):  
Evolution Operator ◽  
Large Time ◽  
Rigid Motion ◽  
Time Dependent ◽  
Energy Relation ◽  
Exterior Domains ◽  
Decay Properties ◽  
Decay Of Solutions ◽  
Whole Space ◽  
Expository Paper

In this expository paper, we study Lq-Lr decay estimates of the evolution operator generated by a perturbed Stokes system in n-dimensional exterior domains when the coefficients are time-dependent and can be unbounded at spatial infinity. By following the approach developed by the present author for the physically relevant case where the rigid motion of the obstacle is time-dependent, we clarify that some decay properties of solutions to the same system in whole space Rn together with the energy relation imply the desired estimates in exterior domains provided n≥3.

Well-Posedness and Uniform Decay Rates for a Nonlinear Damped Schrödinger-Type Equation

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Marcelo M. Cavalcanti ◽  
Valéria N. Domingos Cavalcanti
Keyword(s):  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Differential Operators ◽  
Type Equation ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Dirichlet Boundary Conditions ◽  
Uniform Decay ◽  
Pseudo Differential Operators ◽  
The One

Abstract In this paper we study the existence as well as uniform decay rates of the energy associated with the nonlinear damped Schrödinger equation, i ⁢ u t + Δ ⁢ u + | u | α ⁢ u - g ⁢ ( u t ) = 0   in  ⁢ Ω × ( 0 , ∞ ) , iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty), subject to Dirichlet boundary conditions, where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , n ≤ 3 {n\leq 3} , is a bounded domain with smooth boundary ∂ ⁡ Ω = Γ {\partial\Omega=\Gamma} and α = 2 , 3 {\alpha=2,3} . Our goal is to consider a different approach than the one used in [B. Dehman, P. Gérard and G. Lebeau, Stabilization and control for the nonlinear Schrödinger equation on a compact surface, Math. Z. 254 2006, 4, 729–749], so instead than using the properties of pseudo-differential operators introduced by cited authors, we consider a nonlinear damping, so that we remark that no growth assumptions on g ⁢ ( z ) {g(z)} are made near the origin.

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