Optimal energy decay rates for abstract second order evolution equations with non-autonomous damping

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.

[1]  Jin Liang,et al.  Coupled second order semilinear evolution equations indirectly damped via memory effects , 2013 .

[2]  José A. Langa,et al.  Attractors for infinite-dimensional non-autonomous dynamical systems , 2012 .

[3]  Jun-Ren Luo,et al.  Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping , 2020, Evolution Equations & Control Theory.

[4]  Sakinah,et al.  Vol. , 2020, New Medit.

[5]  Paolo Acquistapace,et al.  Infinite-horizon linear-quadratic regulator problems for nonautonomous parabolic systems with boundary control , 1996 .

[6]  Ramzi May,et al.  Asymptotic for the perturbed heavy ball system with vanishing damping term , 2016 .

[7]  Jin Liang,et al.  APPROXIMATION THEOREMS FOR THE PROPAGATORS OF HIGHER ORDER ABSTRACT CAUCHY PROBLEMS , 2007 .

[8]  N. Burq,et al.  Exponential decay for the damped wave equation in unbounded domains , 2014, 1408.6054.

[9]  José A. Langa,et al.  The stability of attractors for non-autonomous perturbations of gradient-like systems , 2007 .

[10]  A. Haraux,et al.  Optimal decay estimates for the general solution to a class of semi-linear dissipative hyperbolic equations, , 2013, 1306.3644.

[11]  Jerome A. Goldstein,et al.  The non-autonomous wave equation with general Wentzell boundary conditions , 2005, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[12]  Moez Daoulatli.  Rates of decay for the wave systems with time dependent damping , 2011 .

[13]  E. Davies,et al.  SEMIGROUPS OF LINEAR OPERATORS AND APPLICATIONS (Oxford Mathematical Monographs) , 1986 .

[14]  Jin Liang,et al.  Nonautonomous semilinear second order evolution equations with generalized Wentzell boundary conditions , 2012 .

[15]  Jin Liang,et al.  Coupled second order evolution equations with fading memory: Optimal energy decay rate , 2014 .

[16]  S. Nonnenmacher,et al.  Sharp polynomial decay rates for the damped wave equation on the torus , 2012, 1210.6879.

[17]  J. Goldstein.  Semigroups of Linear Operators and Applications , 1985 .

[18]  Alexandre Cabot,et al.  Asymptotics for some semilinear hyperbolic equations with non-autonomous damping , 2012 .

[19]  José A. Langa,et al.  Non-autonomous perturbation of autonomous semilinear differential equations: Continuity of local stable and unstable manifolds , 2007 .

[20]  V. S. Vladimirov.  The Cauchy Problem , 1986 .

[21]  Jacques-Louis Lions.  Contrôlabilite exacte et homogénéisation (I) , 1988 .

[22]  Jin Liang,et al.  Uniform stability of semilinear wave equations with arbitrary local memory effects versus frictional dampings , 2019, Journal of Differential Equations.

[23]  Patrick Martinez,et al.  Precise decay rate estimates for time-dependent dissipative systems , 2000 .

[24]  C. Batty,et al.  L^p-tauberian theorems and L^p-rates for energy decay , 2014, 1403.6084.

[25]  T. Caraballo,et al.  Non-autonomous morse-decomposition and Lyapunov functions for gradient-like processes , 2013 .

[26]  Long time behavior for a semilinear hyperbolic equation with asymtotically vanishing damping term and convex potential , 2014, 1412.7008.

[27]  Jin Liang,et al.  The Cauchy Problem for Higher Order Abstract Differential Equations , 1999 .

[28]  Patrick Martinez,et al.  A new method to obtain decay rate estimates for dissipative systems , 1999 .

[29]  Gabriela Marinoschi,et al.  Degenerate Nonlinear Diffusion Equations , 2012 .

[30]  A. Favini,et al.  An abstract transmission problem in a thin layer, I: Sharp estimates , 2011 .

[31]  D. L. Russell.  Review: J.-L. Lions, Controlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués , 1990 .

[32]  Jerome A. Goldstein,et al.  Time dependent hyperbolic equations , 1969 .

[33]  A. Haraux,et al.  Finding the exact decay rate of all solutions to some second order evolution equations with dissipation , 2014, 1411.6761.

[34]  Degenerate and singular evolution equations in Banach space , 1985 .

[35]  A. Haraux.  Slow and fast decay of solutions to some second order evolution equations , 2005 .

[36]  Alain Haraux.  Decay rate of the range component of solutions to some semilinear evolution equations , 2006 .