2002, 8(3): 675-695. doi: 10.3934/dcds.2002.8.675

Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation

1. 

Departamento de Matemática - Universidade Estadual de Maringá, 87020-900 Maringá - PR, Brazil

Received  June 2001 Revised  October 2001 Published  April 2002

The linear Euler-Bernoulli viscoelastic equation

$u_{t t} +\Delta^2 u-\int_0^t g(t-\tau) \Delta^2 u(\tau)d\tau = 0\quad$ in $\Omega \times (0,\infty)$

subject to nonlinear boundary conditions is considered. We prove existence and uniform decay rates of the energy by assuming a nonlinear and nonlocal feedback acting on the boundary and provided that the kernel of the memory decays exponentially.

Citation: Marcelo Moreira Cavalcanti. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation with nonlocal boundary dissipation. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 675-695. doi: 10.3934/dcds.2002.8.675
[1]

Jong Yeoul Park, Sun Hye Park. On uniform decay for the coupled Euler-Bernoulli viscoelastic system with boundary damping. Discrete & Continuous Dynamical Systems - A, 2005, 12 (3) : 425-436. doi: 10.3934/dcds.2005.12.425

[2]

Denis Mercier. Spectrum analysis of a serially connected Euler-Bernoulli beams problem. Networks & Heterogeneous Media, 2009, 4 (4) : 709-730. doi: 10.3934/nhm.2009.4.709

[3]

Maja Miletić, Dominik Stürzer, Anton Arnold. An Euler-Bernoulli beam with nonlinear damping and a nonlinear spring at the tip. Discrete & Continuous Dynamical Systems - B, 2015, 20 (9) : 3029-3055. doi: 10.3934/dcdsb.2015.20.3029

[4]

Louis Tebou. Well-posedness and stabilization of an Euler-Bernoulli equation with a localized nonlinear dissipation involving the $p$-Laplacian. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2315-2337. doi: 10.3934/dcds.2012.32.2315

[5]

Kaïs Ammari, Denis Mercier, Virginie Régnier, Julie Valein. Spectral analysis and stabilization of a chain of serially connected Euler-Bernoulli beams and strings. Communications on Pure & Applied Analysis, 2012, 11 (2) : 785-807. doi: 10.3934/cpaa.2012.11.785

[6]

Louis Tebou. Energy decay estimates for some weakly coupled Euler-Bernoulli and wave equations with indirect damping mechanisms. Mathematical Control & Related Fields, 2012, 2 (1) : 45-60. doi: 10.3934/mcrf.2012.2.45

[7]

Gen Qi Xu, Siu Pang Yung. Stability and Riesz basis property of a star-shaped network of Euler-Bernoulli beams with joint damping. Networks & Heterogeneous Media, 2008, 3 (4) : 723-747. doi: 10.3934/nhm.2008.3.723

[8]

Fathi Hassine. Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping. Discrete & Continuous Dynamical Systems - B, 2016, 21 (6) : 1757-1774. doi: 10.3934/dcdsb.2016021

[9]

Martin Gugat, Markus Dick. Time-delayed boundary feedback stabilization of the isothermal Euler equations with friction. Mathematical Control & Related Fields, 2011, 1 (4) : 469-491. doi: 10.3934/mcrf.2011.1.469

[10]

Muhammad I. Mustafa. Viscoelastic plate equation with boundary feedback. Evolution Equations & Control Theory, 2017, 6 (2) : 261-276. doi: 10.3934/eect.2017014

[11]

Tobias Breiten, Karl Kunisch. Boundary feedback stabilization of the monodomain equations. Mathematical Control & Related Fields, 2017, 7 (3) : 369-391. doi: 10.3934/mcrf.2017013

[12]

Martin Gugat, Mario Sigalotti. Stars of vibrating strings: Switching boundary feedback stabilization. Networks & Heterogeneous Media, 2010, 5 (2) : 299-314. doi: 10.3934/nhm.2010.5.299

[13]

Kaïs Ammari, Denis Mercier. Boundary feedback stabilization of a chain of serially connected strings. Evolution Equations & Control Theory, 2015, 4 (1) : 1-19. doi: 10.3934/eect.2015.4.1

[14]

Igor Kukavica, Amjad Tuffaha. On the 2D free boundary Euler equation. Evolution Equations & Control Theory, 2012, 1 (2) : 297-314. doi: 10.3934/eect.2012.1.297

[15]

Nguyen H. Sau, Vu N. Phat. LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback. Journal of Industrial & Management Optimization, 2018, 14 (2) : 583-596. doi: 10.3934/jimo.2017061

[16]

Markus Dick, Martin Gugat, Günter Leugering. A strict $H^1$-Lyapunov function and feedback stabilization for the isothermal Euler equations with friction. Numerical Algebra, Control & Optimization, 2011, 1 (2) : 225-244. doi: 10.3934/naco.2011.1.225

[17]

Keng Deng, Zhihua Dong. Blow-up for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 2147-2156. doi: 10.3934/cpaa.2012.11.2147

[18]

Muhammad I. Mustafa. On the control of the wave equation by memory-type boundary condition. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 1179-1192. doi: 10.3934/dcds.2015.35.1179

[19]

Davide Guidetti. Some inverse problems of identification for integrodifferential parabolic systems with a boundary memory term. Discrete & Continuous Dynamical Systems - S, 2015, 8 (4) : 749-756. doi: 10.3934/dcdss.2015.8.749

[20]

Haoyue Cui, Dongyi Liu, Genqi Xu. Asymptotic behavior of a Schrödinger equation under a constrained boundary feedback. Mathematical Control & Related Fields, 2018, 8 (2) : 383-395. doi: 10.3934/mcrf.2018015

2017 Impact Factor: 1.179

Metrics

  • PDF downloads (2)
  • HTML views (0)
  • Cited by (22)

Other articles
by authors

[Back to Top]