We consider an abstract second order non-autonomous evolution equation in a Hilbert space H : u″ + Au + γ(t)u′ + f(u) = 0, where A is a self-adjoint and nonnegative operator on H, f is a conservative H-valued function with polynomial growth (not necessarily to be monotone), and γ(t)u′ is a time-dependent damping term. How exactly the decay of the energy is affected by the damping coefficient γ(t) and the exponent associated with the nonlinear term f? There seems to be little development on the study of such problems, with regard to non-autonomous equations, even for strongly positive operator A. By an idea of asymptotic rate-sharpening (among others), we obtain the optimal decay rate of the energy of the non-autonomous evolution equation in terms of γ(t) and f. As a byproduct, we show the optimality of the energy decay rates obtained previously in the literature when f is a monotone operator.
Keywords: Non-autonomous, abstract second order evolution equation, time dependent damping, energy estimates, slow solutions, nonlinear source, Hilbert space
@article{COCV_2021__27_1_A61_0,
author = {Luo, Jun-Ren and Xiao, Ti-Jun},
title = {Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2021},
publisher = {EDP-Sciences},
volume = {27},
doi = {10.1051/cocv/2021047},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021047/}
}
TY - JOUR AU - Luo, Jun-Ren AU - Xiao, Ti-Jun TI - Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2021 VL - 27 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021047/ DO - 10.1051/cocv/2021047 LA - en ID - COCV_2021__27_1_A61_0 ER -
%0 Journal Article %A Luo, Jun-Ren %A Xiao, Ti-Jun %T Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping %J ESAIM: Control, Optimisation and Calculus of Variations %D 2021 %V 27 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021047/ %R 10.1051/cocv/2021047 %G en %F COCV_2021__27_1_A61_0
Luo, Jun-Ren; Xiao, Ti-Jun. Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping. ESAIM: Control, Optimisation and Calculus of Variations, Tome 27 (2021), article no. 59. doi: 10.1051/cocv/2021047
[1] and , Infinite-horizon linear-quadratic regulator problems for nonautonomous parabolic systems with boundary control. SIAM J. Control Optim. 34 (1996) 1–30.
[2] and , Sharp polynomial decay rates for the damped wave equation on the torus. Anal. PDE 7 (2014) 159–214. (With an appendix by Stéphane Nonnenmacher).
[3] , , and , Non-autonomous Morse decomposition and Lyapunov functions for dynamically gradient processes. Trans. Am. Math. Soc. 365 (2013) 5277–5312.
[4] and , Asymptotic for the perturbed heavy ball system with vanishing damping term. Evol. Equ. Control Theory 6 (2017) 177–186.
[5] , and , L$$-tauberian theorems and L$$-rates for energy decay. J. Functional Anal. 270 (2016) 1153–1201.
[6] and , Exponential decay for the damped wave equation in unbounded domains. Commun. Contemp. Math. 18 (2016) 1650012.
[7] and , Asymptotics for some semilinear hyperbolic equations with non-autonomous damping. J. Differ. Equ. 252 (2012) 294–322.
[8] and , The existence and continuity of stable and unstable manifolds for semilinear problems under non-autonomous perturbation in Banach spaces. J. Differ. Equ. 233 (2007) 622–653.
[9] , and , Attractors for infinite-dimensional non-autonomous dynamical systems. In vol. 182 of Applied Mathematical Sciences. Springer, New York (2013).
[10] , Rates of decay for the wave systems with time-dependent damping. Discrete Contin. Dyn. Syst. 31 (2011) 407–443.
[11] , , and , An abstract transmission problem in a thin layer, I: Sharp estimates. J. Functional Anal. 261 (2011) 1865–1922.
[12] , The Cauchy Problem. With a Foreword by Felix E. Browder, reprint of the 1983 original, In Vol. 18 of Encyclopedia Mathematics and its Applications. Cambridge University Press, Cambridge (2008).
[13] , Degenerate and singular evolution equations in Banach space. Math. Ann. 273 (1985) 17–44.
[14] , , , and , The non-autonomous wave equation with general Wentzell boundary conditions. Proc. Roy. Soc. Edinburgh Sect. A 135 (2005) 317–329.
[15] and , Degenerate Nonlinear Diffusion Equations. Vol. 2049 of Lecture Notes in Mathematics. Springer, Heidelberg (2012).
[16] , Time dependent hyperbolic equations. J. Functional Anal. 4 (1969) 31–49.
[17] , Semigroups of Linear Operators and Applications. Oxford Mathematical Monographs, Oxford University Press, New York (1985).
[18] , and , Optimal decay estimates for the general solution to a class of semil-linear dissipative hyperbolic eqiations. J. Eur. Math. Soc. (JEMS) 18 (2016) 1961–1982.
[19] , and , Finding the exact decay rate of all solutions to some second order evolution equations with dissipation. J. Functional Anal. 271 (2016) 2359–2395.
[20] , Slow and fast decay of solutions to some second order evolution equations. J. Anal. Math. 95 (2005) 297–321.
[21] , Decay rate of the range component of solutions to some semilinear evolution equations. NoDEA Nonlinear Differ. Equ. Appl. 13 (2006) 435–445.
[22] , and , Coupled second order evolution equations with fading memory: Optimal energy decay rate. J. Differ. Equ. 257 (2014) 1501–1528.
[23] , and , Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings. J. Differ. Equ. 266 (2019) 7230–7263.
[24] and , Vol. 176 of Nonautonomous Dynamical Systems. Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2011).
[25] , , and , The stability of attractors for non-autonomous perturbations of gradient-like systems. J. Differ. Equ. 234 (2007) 607–625.
[26] , and , Approximation theorems for the propagators of higher order abstract Cauchy problems. Trans. Am. Math. Soc. 360 (2008) 1723–1739.
[27] , Controlabilité exacte, perturbations et stabilisation de systèmes distribués 1, Masson, Paris (1988).
[28] and , Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evol. Equ. Control Theory 9 (2020) 359–373.
[29] , A new method to obtain decay rate estimates for dissipative systems. ESAIM: COCV 4 (1999) 419–444.
[30] , Precise decay rate estimates for time-dependent dissipative systems. Israel J. Math. 119 (2000) 291–324.
[31] , Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential. J. Math. Anal. Appl. 430 (2015) 410–416.
[32] , On the time decay of solutions of the wave equation with a local time-dependent nonlinear dissipation. Adv. Math. Sci. Appl. 7 (1997) 317–331.
[33] and , Coupled second order semilinear evolution equations indirectly damped via memory effects. J. Differ. Equ. 254 (2013) 2128–2157.
[34] and , Nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions. J. Differ. Equ. 252 (2012) 3953–3971.
[35] and , Vol. 1701 of The Cauchy Problem for Higher Order Abstract Differential Equations. Lecture Notes in Mathematics. Springer, Berlin (1998).
Cité par Sources :
The work was supported partly by the NSF of China (11771091, 11831011), and the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900).
