Asymptotic behavior of weakly dissipative Bresse‐Timoshenko system on influence of the second spectrum of frequency

Abstract

In the present work, we consider a dissipative Bresse‐Timoshenko type system, which is free of physical anomaly know as second spectrum of frequency according important observations made by Elishakoff et al., and we establish a new result of exponential decay. We prove that frictional damping acting on vertical displacement of this system is exponentially stable regardless the equality between velocities of wave propagation. This result is new and completely different from ones obtained by Almeida Júnior et al. for the Timoshenko system with the same damping acting on vertical displacement. Our approach is strongly inspired in a recent result due to Almeida Júnior and Ramos where the authors justified, from physical point of view, a classical and pioneering result due to Soufyane. That is, the dissipative mechanism of frictional type acting on angular rotation of the Timoshenko system truncates the consequences of the second spectrum when the velocities of wave propagation are equal. Also, for dissipative Bresse‐Timoshenko systems which are free of the second spectrum, the exponential decay occurs regardless of any condition on coefficients of system. Based on the new results, it is demonstrated that the classical Bresse‐Timoshenko beam theory, as an overcomplicated theory, requires damping terms to eliminate the second spectrum.

A new scenario for stability of nonlinear Bresse‐Timoshenko type systems with time dependent delay

“…For respective proofs, see Almeida Júnior et al [3,4] Regarding to dissipative Timoshenko type systems with time delay terms, we point out the work due to Said-Hoauri and Larskri. [37] To the best of our knowledge, these authors were the first one for contributing in literature for dissipative Timoshenko systems by taking into account a delay constant term.…”

“…In this paper, we deal with a nonlinear Bresse-Timoshenko type system with time dependent delay term and we study the exponential decay of the total energy based on the recent works by Almeida Júnior et al [3][4][5] These authors bring to light the problem of the damage consequences of the so called second spectrum of frequencies, or simply second spectrum, in the context of the stabilization of dissipative models of the Timoshenko type. In particular, the main contributions made by Almeida Júnior and Ramos [3] were twofold.…”

Bresse-Timoshenko type systems with thermodiffusion effects: Well-possedness, stability and numerical results

“…Despite the fact that a sufficient number of works have been devoted to the study of natural vibrations of a Breese-Timoshenko beam, the problem of determining qualitative properties with thermal, mass diffusion and theormoelastic effects remains unsolved. [2,3,5,10,13,19] In [8], the authors studied stability of thermoviscoelastic Bresse beam system. The exponential decay of energy is proved and implicit Euler type scheme based on finite differences in time and finite elements in spaces is introduced to show that the discrete energy decreases in time and the author obtained an error estimates.…”

“…In both systems (1.4) and (1.5), the authors used an appropriate Lyapunov functional to prove an exponential decay results, regardless of any relationship between wave propagation velocities. (See [1,2,3,19]). The present article is a logical continuation of works [7,10,13].…”

Asymptotic behavior of the porous elastic system with dual phase lag model: Classical versus second spectrum perspectives

“…They further show that for any values of the system's coefficients, dissipative Timoshenko-type systems free of the second spectrum are exponentially stable, based on crucial scientific and historical discoveries by Elishakoff. For more results about the stability of the Bresse-Timoshenko type systems see the following articles. 18,[19][20][21] The rest of this paper is as follows: In Section 2, we study the well-posedness of the onedimensional porous materials with microtemperature. In Section 3, we establish the exponential decay under the conditions on the one-dimensional porous materials with microtemperature parameters.…”