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  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 & 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.  Google Scholar

[2]

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, 108. North-Holland Publishing Co., Amsterdam, 1985.  Google Scholar

[3]

M. GhisiM. 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.  Google Scholar

[4]

M. GhisiM. 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.  Google Scholar

[5]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. HarauxP. 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.  Google Scholar

[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.  Google Scholar

[11]

K.-P. JinJ. 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.  Google Scholar

[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.  Google Scholar

[13]

P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Israel J. Math., 119 (2000), 291-324.  doi: 10.1007/BF02810672.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

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.  Google Scholar

[2]

H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, North-Holland Mathematics Studies, 108. North-Holland Publishing Co., Amsterdam, 1985.  Google Scholar

[3]

M. GhisiM. 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.  Google Scholar

[4]

M. GhisiM. 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.  Google Scholar

[5]

J. A. Goldstein, Semigroups of Linear Operators and Applications, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York, 1985.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

A. HarauxP. 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.  Google Scholar

[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.  Google Scholar

[11]

K.-P. JinJ. 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.  Google Scholar

[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.  Google Scholar

[13]

P. Martinez, Precise decay rate estimates for time-dependent dissipative systems, Israel J. Math., 119 (2000), 291-324.  doi: 10.1007/BF02810672.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[2]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[3]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[4]

Thomas Bartsch, Tian Xu. Strongly localized semiclassical states for nonlinear Dirac equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 29-60. doi: 10.3934/dcds.2020297

[5]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[6]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[7]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450

[8]

Weisong Dong, Chang Li. Second order estimates for complex Hessian equations on Hermitian manifolds. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020377

[9]

Noriyoshi Fukaya. Uniqueness and nondegeneracy of ground states for nonlinear Schrödinger equations with attractive inverse-power potential. Communications on Pure & Applied Analysis, 2021, 20 (1) : 121-143. doi: 10.3934/cpaa.2020260

[10]

Hassan Mohammad. A diagonal PRP-type projection method for convex constrained nonlinear monotone equations. Journal of Industrial & Management Optimization, 2021, 17 (1) : 101-116. doi: 10.3934/jimo.2019101

[11]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

[12]

Kihoon Seong. Low regularity a priori estimates for the fourth order cubic nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5437-5473. doi: 10.3934/cpaa.2020247

[13]

Nguyen Thi Kim Son, Nguyen Phuong Dong, Le Hoang Son, Alireza Khastan, Hoang Viet Long. Complete controllability for a class of fractional evolution equations with uncertainty. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020104

[14]

Mostafa Mbekhta. Representation and approximation of the polar factor of an operator on a Hilbert space. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020463

[15]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[16]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[17]

Predrag S. Stanimirović, Branislav Ivanov, Haifeng Ma, Dijana Mosić. A survey of gradient methods for solving nonlinear optimization. Electronic Research Archive, 2020, 28 (4) : 1573-1624. doi: 10.3934/era.2020115

[18]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366

[19]

D. R. Michiel Renger, Johannes Zimmer. Orthogonality of fluxes in general nonlinear reaction networks. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 205-217. doi: 10.3934/dcdss.2020346

[20]

Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137

2019 Impact Factor: 0.953

Metrics

  • PDF downloads (233)
  • HTML views (430)
  • Cited by (1)

Other articles
by authors

[Back to Top]