DECAY PROPERTY FOR THE TIMOSHENKO SYSTEM WITH MEMORY-TYPE DISSIPATION

2012 ◽  
Vol 22 (02) ◽  
pp. 1150012 ◽  
Author(s):  
YONGQIN LIU ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Decay Estimate ◽  
Pointwise Estimate ◽  
Decay Property ◽  
Memory Term ◽  
Fourier Space ◽  
Initial Value ◽  
Timoshenko System ◽  
Frictional Dissipation ◽  
Memory Type ◽  
With Memory

In this paper we consider the initial value problem for the Timoshenko system with a memory term. We construct the fundamental solution by using the Fourier–Laplace transform and obtain the solution formula of the problem. Moreover, applying the energy method in the Fourier space, we derive the pointwise estimate of solutions in the Fourier space, which gives a sharp decay estimate of solutions. It is shown that the decay property of the system is of the regularity-loss type and is weaker than that of the Timoshenko system with a frictional dissipation.

Decay property of the Timoshenko–Cattaneo system

2016 ◽  
Vol 14 (03) ◽  
pp. 393-413 ◽  
Author(s):  
Naofumi Mori ◽  
Shuichi Kawashima
Keyword(s):  
Heat Conduction ◽  
Decay Estimate ◽  
Low Frequency ◽  
Pointwise Estimate ◽  
Frequency Region ◽  
Decay Property ◽  
General Situation ◽  
High Frequency Region ◽  
Timoshenko System ◽  
Whole Space

We study the Timoshenko system with Cattaneo’s type heat conduction in the one-dimensional whole space. We investigate the dissipative structure of the system and derive the optimal [Formula: see text] decay estimate of the solution in a general situation. Our decay estimate is based on the detailed pointwise estimate of the solution in the Fourier space. We observe that the decay property of our Timoshenko–Cattaneo system is of the regularity-loss type. This decay property is a little different from that of the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]) in the low frequency region. However, in the high frequency region, it is just the same as that of the Timoshenko–Fourier system (see [N. Mori and S. Kawashima, Decay property for the Timoshenko system with Fourier’s type heat conduction, J. Hyperbolic Differential Equations 11 (2014) 135–157]) or the dissipative Timoshenko system (see [K. Ide, K. Haramoto and S. Kawashima, Decay property of regularity-loss type for dissipative Timoshenko system, Math. Models Methods Appl. Sci. 18 (2008) 647–667]), although the stability number is different. Finally, we study the decay property of the Timoshenko system with the thermal effect of memory-type by reducing it to the Timoshenko–Cattaneo system.

DECAY PROPERTY OF REGULARITY-LOSS TYPE AND NONLINEAR EFFECTS FOR DISSIPATIVE TIMOSHENKO SYSTEM

2008 ◽  
Vol 18 (07) ◽  
pp. 1001-1025 ◽  
Author(s):  
KENTARO IDE ◽  
SHUICHI KAWASHIMA
Keyword(s):  
Nonlinear Effects ◽  
Decay Property ◽  
Regularity Conditions ◽  
Initial Value ◽  
Timoshenko System ◽  
Linear Diffusion ◽  
Diffusion Wave ◽  
Decay Of Solutions ◽  
Nonlinear Version ◽  
Derivatives Of

We consider the initial value problem for a nonlinear version of the dissipative Timoshenko system. This syetem verifies the decay property of regularity-loss type. To overcome this difficulty caused by the regularity-loss property, we employ the time weighed L2energy method which is combined with the optimal L2decay estimates for lower order derivatives of solutions. Then we show the global existence and asymptotic decay of solutions under smallness and enough regularity conditions on the initial data. Moreover, we show that the solution approaches the linear diffusion wave expressed in terms of the superposition of the heat kernels as time tends to infinity.

scholarly journals Decay property for a plate equation with memory-type dissipation

2011 ◽  
Vol 4 (2) ◽  
pp. 531-547 ◽  
Author(s):  
Yongqin Liu ◽  
◽  
Shuichi Kawashima ◽  
Keyword(s):  
Decay Property ◽  
Plate Equation ◽  
Memory Type ◽  
With Memory

scholarly journals PULLBACK ATTRACTORS FOR A NONAUTONOMOUS INTEGRO-DIFFERENTIAL EQUATION WITH MEMORY IN SOME UNBOUNDED DOMAINS

2013 ◽  
Vol 23 (03) ◽  
pp. 1350042
Author(s):  
MARÍA ANGUIANO ◽  
TOMÁS CARABALLO ◽  
JOSÉ REAL ◽  
JOSÉ VALERO
Keyword(s):  
Unbounded Domains ◽  
Memory Term ◽  
Pullback Attractors ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Diffusion Type ◽  
Nonautonomous Dynamical Systems ◽  
Infinite Delays ◽  
Convolution Integrals ◽  
With Memory

The main aim of this paper is to analyze the asymptotic behavior of a nonautonomous integro-differential parabolic equation of diffusion type with a memory term, expressed by convolution integrals involving infinite delays, in an unbounded domain. The assumptions imposed do not ensure uniqueness of solutions of the corresponding initial value problems. The theory of set-valued nonautonomous dynamical systems is applied to prove the existence of pullback attractors for our model. To do this, we first analyze an abstract version of the equation.

scholarly journals A New General Decay Rate of Wave Equation with Memory-Type Boundary Control

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Sheng Fan
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
General Decay ◽  
Forced Oscillations ◽  
Memory Term ◽  
General Decay Rate ◽  
Damped Oscillations ◽  
Memory Type ◽  
Type Boundary ◽  
With Memory

Of interest is a wave equation with memory-type boundary oscillations, in which the forced oscillations of the rod is given by a memory term at the boundary. We establish a new general decay rate to the system. And it possesses the character of damped oscillations and tends to a finite value for a large time. By assuming the resolvent kernel that is more general than those in previous papers, we establish a more general energy decay result. Hence the result improves earlier results in the literature.

scholarly journals Exponential Attractors for the 3D Fractional-order Bardina Turbulence Model With Memory and Horizontal Filtering

2021 ◽  
Author(s):  
Michele Annese ◽  
Luca Bisconti ◽  
Davide Catania
Keyword(s):  
Phase Space ◽  
Fractional Order ◽  
Turbulence Model ◽  
Memory Term ◽  
Exponential Attractor ◽  
Exponential Attractors ◽  
Regularity Properties ◽  
Hereditary Effects ◽  
With Memory ◽  
Simplified Bardina

AbstractWe consider the 3D simplified Bardina turbulence model with horizontal filtering, fractional dissipation, and the presence of a memory term incorporating hereditary effects. We analyze the regularity properties and the dissipative nature of the considered system and, in our main result, we show the existence of a global exponential attractor in a suitable phase space.

Optimal L 2 -decay of solutions to a cubic dissipative nonlinear Schrödinger equation

2021 ◽  
pp. 1-13
Author(s):  
Kita Naoyasu ◽  
Sato Takuya
Keyword(s):  
Sobolev Space ◽  
Global Solution ◽  
Initial Value Problem ◽  
Trivial Solution ◽  
Decay Estimate ◽  
Weighted Sobolev Space ◽  
The Other ◽  
Initial Value ◽  
Decay Of Solutions ◽  
Dissipative Nonlinearity

This paper presents the optimality of decay estimate of solutions to the initial value problem of 1D Schrödinger equations containing a long-range dissipative nonlinearity, i.e., λ | u | 2 u. Our aim is to obtain the two results. One asserts that, if the L 2 -norm of a global solution, with an initial datum in the weighted Sobolev space, decays at the rate more rapid than ( log t ) − 1 / 2 , then it must be a trivial solution. The other asserts that there exists a solution decaying just at the rate of ( log t ) − 1 / 2 in L 2 .

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