American Institute of Mathematical Sciences

June  2020, 9(2): 359-373. doi: 10.3934/eect.2020009

Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping

 Shanghai Key Laboratory for Contemporary Applied Mathematics, School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Corresponding author: Ti-Jun Xiao

Received  January 2019 Revised  April 2019 Published  June 2020 Early access  August 2019

Fund Project: The work was supported partly by the NSF of China (11771091, 11831011), the Fudan University (IDH 1411016), and the Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900)

The paper is concerned with the Cauchy problem for second order hyperbolic evolution equations with nonlinear source in a Hilbert space, under the effect of nonlinear time-dependent damping. With the help of the method of weighted energy integral, we obtain explicit decay rate estimates for the solutions of the equation in terms of the damping coefficient and two nonlinear exponents. Specialized to the case of linear, time-independent damping, we recover the corresponding decay rates originally obtained in [3] via a different way. Moreover, examples are given to show how to apply our abstract results to concrete problems concerning damped wave equations, integro-differential damped equations, as well as damped plate equations.

Citation: Jun-Ren Luo, Ti-Jun Xiao. Decay rates for second order evolution equations in Hilbert spaces with nonlinear time-dependent damping. Evolution Equations and Control Theory, 2020, 9 (2) : 359-373. doi: 10.3934/eect.2020009
References:
 [1] M. Daoulatli, Rates of decay for the wave systems with time-dependent damping, Discrete Contin. Dyn. Syst., 31 (2011), 407-443.  doi: 10.3934/dcds.2011.31.407. [2] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, 108. North-Holland Publishing Co., Amsterdam, 1985. [3] M. Ghisi, M. Gobbino and A. Haraux, 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.  doi: 10.4171/JEMS/635. [4] M. Ghisi, M. Gobbino and A. Haraux, Finding the exact decay rate of all solutions to some second order evolution equations with dissipation, J. Funct. Anal., 271 (2016), 2359-2395.  doi: 10.1016/j.jfa.2016.08.010. [5] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985. [6] A. Haraux, Slow and fast decay of solutions to some second order evolution equations, J. Anal. Math., 95 (2005), 297-321.  doi: 10.1007/BF02791505. [7] A. Haraux and M. A. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evolution Equations and Control Theory, 2 (2013), 461-470.  doi: 10.3934/eect.2013.2.461. [8] A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems, Classical Methods and Recent Advances, BCAM SpringerBriefs. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2015. doi: 10.1007/978-3-319-23407-6. [9] A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations, SIAM J. Control Optim., 43 (2005), 2089-2108.  doi: 10.1137/S0363012903436569. [10] Z. Jiao and T.-J. Xiao, Convergence and speed estimates for semilinear wave systems with nonautonomous damping, Math. Methods Appl. Sci., 39 (2016), 5465-5474.  doi: 10.1002/mma.3931. [11] K.-P. Jin, J. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differential Equations, 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018. [12] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116. [13] P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Israel J. Math., 119 (2000), 291-324.  doi: 10.1007/BF02810672. [14] R. May, Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential, J. Math. Anal. Appl., 430 (2015), 410-416.  doi: 10.1016/j.jmaa.2015.04.067. [15] M. Nakao, 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. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [17] T.-J. Xiao and J. Liang, Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differential Equations, 254 (2013), 2128-2157.  doi: 10.1016/j.jde.2012.11.019.

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References:
 [1] M. Daoulatli, Rates of decay for the wave systems with time-dependent damping, Discrete Contin. Dyn. Syst., 31 (2011), 407-443.  doi: 10.3934/dcds.2011.31.407. [2] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, 108. North-Holland Publishing Co., Amsterdam, 1985. [3] M. Ghisi, M. Gobbino and A. Haraux, 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.  doi: 10.4171/JEMS/635. [4] M. Ghisi, M. Gobbino and A. Haraux, Finding the exact decay rate of all solutions to some second order evolution equations with dissipation, J. Funct. Anal., 271 (2016), 2359-2395.  doi: 10.1016/j.jfa.2016.08.010. [5] J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985. [6] A. Haraux, Slow and fast decay of solutions to some second order evolution equations, J. Anal. Math., 95 (2005), 297-321.  doi: 10.1007/BF02791505. [7] A. Haraux and M. A. Jendoubi, Asymptotics for a second order differential equation with a linear, slowly time-decaying damping term, Evolution Equations and Control Theory, 2 (2013), 461-470.  doi: 10.3934/eect.2013.2.461. [8] A. Haraux and M. A. Jendoubi, The Convergence Problem for Dissipative Autonomous Systems, Classical Methods and Recent Advances, BCAM SpringerBriefs. Springer, Cham; BCAM Basque Center for Applied Mathematics, Bilbao, 2015. doi: 10.1007/978-3-319-23407-6. [9] A. Haraux, P. Martinez and J. Vancostenoble, Asymptotic stability for intermittently controlled second-order evolution equations, SIAM J. Control Optim., 43 (2005), 2089-2108.  doi: 10.1137/S0363012903436569. [10] Z. Jiao and T.-J. Xiao, Convergence and speed estimates for semilinear wave systems with nonautonomous damping, Math. Methods Appl. Sci., 39 (2016), 5465-5474.  doi: 10.1002/mma.3931. [11] K.-P. Jin, J. Liang and T.-J. Xiao, Coupled second order evolution equations with fading memory: Optimal energy decay rate, J. Differential Equations, 257 (2014), 1501-1528.  doi: 10.1016/j.jde.2014.05.018. [12] P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM Control Optim. Calc. Var., 4 (1999), 419-444.  doi: 10.1051/cocv:1999116. [13] P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Israel J. Math., 119 (2000), 291-324.  doi: 10.1007/BF02810672. [14] R. May, Long time behavior for a semilinear hyperbolic equation with asymptotically vanishing damping term and convex potential, J. Math. Anal. Appl., 430 (2015), 410-416.  doi: 10.1016/j.jmaa.2015.04.067. [15] M. Nakao, 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. [16] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1. [17] T.-J. Xiao and J. Liang, Coupled second order semilinear evolution equations indirectly damped via memory effects, J. Differential Equations, 254 (2013), 2128-2157.  doi: 10.1016/j.jde.2012.11.019.
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