Dissipative Timoshenko System Research Articles

Issues related to the second spectrum, Ostrogradsky’s energy and the stabilization of Timoshenko–Ehrenfest-type systems

In this paper, we discuss the stabilization properties of a beam model on a Winkler foundation by using Timoshenko–Ehrenfest-type systems, taking into account the influence of the so-called second spectrum. We consider the well-known classical version of the Timoshenko–Ehrenfest beam model as well as the truncated (or simplified) version of the same beam model according to the approach given by Elishakoff (in: Banks-Sills (ed.), Advances in mathematical modelling and experimental methods for materials and structures, solid mechanics and its applications. Springer, Berlin, pp 249–254, 2010). The main novelty of our approach is the concept of applying Ostrogradsky’s energy to both beam models to highlight the physics issues arising in the frequency spectra. Our ideas are an attempt to fill the gap regarding the consequences of the second spectrum in the stabilization scenario for dissipative Timoshenko systems that are partially damped.

The hypothesis of equal wave speeds for stabilization of Timoshenko beam is not necessary anymore: the time delay cases†

AbstractIn the current study, we consider the Bresse–Timoshenko type systems and we prove some stability results for time delay cases into setting of so called simplified Bresse–Timoshenko equations (or truncated version of Bresse–Timoshenko equations) according to contributions of Elishakoff et al. (2010, Advances in Mathematical Modeling and Experimental Methods for Materials and Structures. Solid Mechanics and Its Applications. Springer: Berlin, 249–254.; 2015, Celebrating the Centenary of Timoshenko’s study of effects of shear deformation and rotary inertia. Appl. Mech. Rev.67, 1–11.; 2017, Critical contrasting of three versions of vibrating Bresse-Timoshenko beam with a crack. Int. J. Solids Struct. 109, 143–151.). These equations are free of the so-called ‘second spectrum’ phenomenon, and they have important consequences on stabilization setting. Specifically, following Almeida Júnior and Ramos (2017, On the nature of dissipative Timoshenko systems at light of the second spectrum. Z. Angew. Math. Phys.68, 31.) in a recent contribution that shows that damping effects eliminate the consequences of this spectrum for equal wave propagation velocities, we prove that time delay effects are able of stabilizing the truncated version regardless of any relationship between coefficients of system. It is concluded that dissipative truncated versions of Bresse–Timoshenko equations are advantageous over the classical Bresse–Timoshenko equations in stabilization context.

On the nature of dissipative Timoshenko systems at light of the second spectrum of frequency

In the present work, we prove that there exists a relation between a physical inconsistence known as second spectrum of frequency or non-physical spectrum and the exponential decay of a dissipative Timoshenko system where the damping mechanism acts on angle rotation. The so-called second spectrum is addressed into stabilization scenario and, in particular, we show that the second spectrum of the classical Timoshenko model can be truncated by taking a damping mechanism. Also, we show that dissipative Timoshenko type systems which are free of the second spectrum [based on important physical and historical observations made by Elishakoff (Advances mathematical modeling and experimental methods for materials and structures, solid mechanics and its applications, Springer, Berlin, pp 249–254, 2010), Elishakoff et al. (ASME Am Soc Mech Eng Appl Mech Rev 67(6):1–11 2015) and Elishakoff et al. (Int J Solids Struct 109:143–151, 2017)] are exponential stable for any values of the coefficients of system. In this direction, we provide physical explanations why weakly dissipative Timoshenko systems decay exponentially according to equality between velocity of wave propagation as proved in pioneering works by Soufyane (C R Acad Sci 328(8):731–734, 1999) and also by Munoz Rivera and Racke (Discrete Contin Dyn Syst B 9:1625–1639, 2003). Therefore, the second spectrum of the classical Timoshenko beam model plays an important role in explaining some results on exponential decay and our investigations suggest to pay attention to the eventual consequences of this spectrum on stabilization setting for dissipative Timoshenko type systems.

Decay property of the Timoshenko–Cattaneo system

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.

Uniform decay in weakly dissipative timoshenko system with internal distributed delay feedbacks

In this paper we consider one-dimensional Timoshenko system with linear frictional damping and a distributed delay acting on the displacement equation. Under suitable assumptions on the weight of the delay and the wave speeds, we establish the well-posedness of the system and show that the dissipation through the frictional damping is strong enough to uniformly stabilize the system even in the presence of delay.

Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks

In this paper, we consider a one-dimensional Timoshenko system with a linear frictional damping and a constant delay acting on the displacement equation. Under appropriate assumptions on the weight of the delay and the wave speeds, we establish the well posedness and asymptotic stability results of the system. Our stability results show that the dissipation through the frictional damping is strong enough to uniformly stabilize the system even in the presence of delay.

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

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.

Stability to weakly dissipative Timoshenko systems

In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.

Decay rates and global existence for semilinear dissipative Timoshenko systems

We prove new decay estimates for the dissipative Timoshenko system in the one-dimensional whole space, and a global existence theorem for semilinear systems. More precisely, if we restrict the initial data ( ( φ 0 , ψ 0 ) , ( φ 1 , ψ 1 ) ) ( (\varphi _{0},\psi _0),(\varphi _{1},\psi _1)) ∈ ( H s + 1 ( R N ) ∩ L 1 , γ ( R N ) ) × ( H s ( R N ) ) ∩ L 1 , γ ( R N ) ) \in \Big ( H^{s+1}( \mathbb {R}^{N}) \cap L^{1,\gamma }( \mathbb {R}^{N}) \Big ) \times \Big ( H^{s}( \mathbb {R}^{N})\Big ) \cap L^{1,\gamma }( \mathbb {R}^{N}) \Big ) with γ ∈ [ 0 , 1 ] \gamma \in \left [ 0,1\right ] , then we can derive faster decay estimates than those given by Ide, Haramoto and Kawashima. In addition, we use these decay estimates of the linear problem combined with the weighted energy method introduced by Todorova and Yordanov with the special weight given by Ikehata and Inoue to solve a semilinear problem with low regularity.

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

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 L2 energy method which is combined with the optimal L2 decay 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.

DECAY PROPERTY OF REGULARITY-LOSS TYPE FOR DISSIPATIVE TIMOSHENKO SYSTEM

We study the decay property of the dissipative Timoshenko system in the one-dimensional whole space. We derive the L2 decay estimates of solutions in a general situation and observe that this decay structure is of the regularity-loss type. Also, we give a refinement of these decay estimates for some special initial data. Moreover, under enough regularity assumption on the initial data, we show that the solution approaches the linear diffusion wave expressed in terms of the heat kernels as time tends to infinity. The proof is based on the detailed pointwise estimates of solutions in the Fourier space.

DECAY PROPERTY OF REGULARITY-LOSS TYPE AND APPLICATION TO SOME NONLINEAR HYPERBOLIC–ELLIPTIC SYSTEM

We discuss the global solvability and asymptotic behavior of solutions to the Cauchy problem for some nonlinear hyperbolic–elliptic system with a fourth-order elliptic part. This system is a modified version of the simplest radiating gas model and verifies a decay property of regularity-loss type. Such a dissipative structure also appears in the dissipative Timoshenko system studied by Rivera and Racke. This dissipative property is very weak in high frequency region and causes the difficulty in deriving the desired a priori estimates for global solutions to the nonlinear problem. In fact, it turns out that the usual energy method does not work well. We overcome this difficulty by employing a time-weighted energy method which is combined with the optimal decay for lower order derivatives of solutions, and we establish a global existence and asymptotic decay result. Furthermore, we show that the solution has an asymptotic self-similar profile described by the Burgers equation as time tends to infinity.