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

• Published in: Zeitschrift für angewandte Mathematik und Mechanik Vol. 100; no. 2
• Main Authors: Feng, B., Almeida, D. S., Santos, M. J., Rosário Miranda, L. G.
• Format: Journal Article
• Language: English
• Published: Weinheim Wiley Subscription Services, Inc 01.02.2020
• Subjects: 35B35; 35B40; 35Q74; 74D05; 74K10; Asymptotic properties; Bresse‐Timoshenko‐type systems; Delay; exponential decay; Nonlinear systems; phase velocities; Propagation velocity; second spectrum; Stability; Time dependence; time dependent delay; Viscous damping; Wave propagation
• ISSN: 0044-2267; 1521-4001
• DOI: 10.1002/zamm.201900160

In this paper, we study the asymptotic behavior for a Bresse‐Timoshenko type system with time‐dependent delay terms. Our results follow a recent approach given by Almeida Júnior and Ramos (Zeitschrift für angewandte Mathematik und Physik. 68, 1‐31. 2017) where they showed that the viscous damping acting on angle rotation of the classical Bresse‐Timoshenko model eliminates the damage consequences of the so called second spectrum of frequency. Moreover, for Bresse‐Timoshenko type systems which are free of these consequences the assumption of equal wave speeds is not necessary anymore to get the exponential stability. Here, by introducing an appropriate Lyapunov functional we prove the exponential stability for time dependent delay cases regardless of any relationship between wave propagation velocities. In this paper, we study the asymptotic behavior for a Bresse‐Timoshenko type system with time‐dependent delay terms. Our results follow a recent approach given by Almeida Júnior and Ramos (Zeitschrift für angewandte Mathematik und Physik. 68, 1‐31. 2017) where they showed that the viscous damping acting on angle rotation of the classical Bresse‐Timoshenko model eliminates the damage consequences of the so called second spectrum of frequency….