Abstract
This work is devoted to prove the polynomial decay for the energy of solutions of a nonlinear plate equation of Bernoulli-Euler type with a nonlinear localized damping term. Following the methods in [20], which combines energy estimates, multipliers and compactness arguments the problem is reduced to a unique continuation question. In [24] the case where the damping is linear was solved. In this article we address the general case and obtain explicit rates of decay that depend on the growth of the dissipative term near zero and infinity.
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References
F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl Math. Optim. 51 (2005) 61–105.
M.A. Astaburuaga and R.C. Charão, Stabilization of the total energy for a system of elasticity with localized dissipation, Diff. Int. Equations 15(11) (2002) 1357–1376.
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM J. Control and Opt. 30 (1992) 1024–1065.
F. Conrad and B. Rao, Decay of solutions of wave equations in a star-shaped domain with nonlinear boundary feedback, Asymptotic Anal. 7 (1993) 159–177.
C.D. Dafermos, On the existence and the asymptotic stability of solutions to the equations of linear thermoelasticity, Arch. Rational Mech. Anal. 29 (1968) 241–271.
A. Haraux, Stabilization of trajectories for some weakly damped hyperbolic equations, J. Diff. Equations 59 (1985) 145–154.
A. Haraux, Oscillations forcées pour certains système dissipatifs nonlinéaires, Publications du Laboratoire d’Analyse Numérique No 78010, Université Pierre et Marie Curie (1978).
A. Haraux, Semi-groupes linéaires equation d’évolution linéaires périodiques, Publications du Laboratoire d’Analyse Numérique No 78011, Université Pierre et Marie Curie (1978).
M.A. Horn, Nonlinear boundary stabilization of a system of anisotropic elasticity with light internal damping, Contemporary Mathematics 268 (2000), 177–189.
J.U. Kim, Exact semi-internal controllability of an Euler-Bernoulli equation, SIAM J. Control and Opt. 30 (1992) 1001–1023.
V. Komornik, Exact controllabitily in ans Stabilization. The multiplier Method, Masson, Paris and Wiley, New York (1994).
I. Lasiecka and D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Diff. Int. Equations 6 (1993), 507–533.
J.L. Lions, Contrôlabilité exacte, perturbations et stabilization de systèmes distribués, Tome 1, Paris, Masson (1988).
P. Martinez, Boundary Stabilization of the Wave Equation in almost star-shaped domains, SIAM J. Control Optim. 37(3)(1999), 673–694.
P. Martinez, A new method to obtain decay estimates for dissipative systems with localized damping, Rev. Mat. Complut. 12(1)(1999), 251–283.
G. Perla Menzala, A.F. Pazoto and E. Zuazua, Stabilization of Berger-Timoshenko’s equation as limit of the uniform stabilization of the von Kármán system of beams and plates, M2AN. Mathematical Modelling and Numerical Analysis 36(4) (2002), 657–691.
G. Perla Menzala and E. Zuazua, Timoshenko’s beam equation as a limit of a nonliner one-dimensional von Kármán system, Proc. Royal Soc. Edinburgh, Sect. A 130 (2000), 855–875.
G. Perla Menzala and E. Zuazua, Timoshenko’s plate equation as singular limit of the dynamical von Kármán system, J. Math. Pures Appl. 79(1) (2000), 73–94.
G. Perla Menzala and E. Zuazua, Explicit exponential decay rates for solutions of von Kármán system of thermoelastic plates, C. R. Acad. Sci. Paris 324 Série I (1997), 49–54.
M. Nakao, Decay of solutions of the wave equation with a local nonlinear dissipation, Math. Ann. 305(3) (1996), 403–417.
M. Nakao, A difference inequality and application to nonlinear evolution equations, J. Math. Soc. Japan, 30 (1978) 747–762.
L.R. Roder Tcheugoué, Stabilization of the wave equation with localized nonlinear damping, J. Diff. Equations 145 (1998), 502–524.
M. Slemrod, Weak asymptotic decay via a “Relaxed Invariance Principle” for a wave equation with nonlinear, nonmonotone damping, Proc. Royal Soc. Edinburgh 113 (1989), 87–97.
M. Tucsnak, Semi-internal stabilization for a non-linear Bernoulli-Euler equation, Math. Methods Appl. Sci. 19(1) (1996), 897–907.
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. in PDE 15 (1990), 205–235.
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Charão, R., Bisognin, E., Bisognin, V., Pazoto, A. (2005). Asymptotic Behavior of a Bernoulli-Euler Type Equation with Nonlinear Localized Damping. In: Cazenave, T., et al. Contributions to Nonlinear Analysis. Progress in Nonlinear Differential Equations and Their Applications, vol 66. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7401-2_5
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DOI: https://doi.org/10.1007/3-7643-7401-2_5
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