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doi: 10.3934/eect.2020086

Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation

Tomomi Yokota , and  Kentarou Yoshii

Department of Mathematics, Tokyo University of Science, 1-3, Kagurazaka, Shinjuku-ku, Tokyo 162-8601, Japan

* Corresponding author: Tomomi Yokota

Received  February 2020 Revised  June 2020 Early access  August 2020

Fund Project: The first author is supported by Grant-in-Aid for Scientific Research (C), No. 16K05182

This paper deals with the solvability in the semilinear abstract evolution equation with countable time delays,
$ \begin{equation*} \begin{cases} \dfrac{du}{dt}(t)+Au(t) = F(u(t), (u(t-\tau_n))_{n\in\mathbb{N}}), & t>0, \\ u(t) = u_0(t), & t \in \bigcup\limits_{n \in \mathbb{N}}[-\tau_n,0], \end{cases} \end{equation*} $
in a Banach space
$ X $
, where
$ -A $
 is a generator of a
$ C_0 $
-semigroup with exponential decay and
$ F: X \times X^\mathbb{N} \to X $
 is Lipschitz continuous. Nicaise and Pignotti (J. Evol. Equ.; 2018;18;947–971) established global existence and exponential decay in time for solutions of the above equation with finite time delays in Hilbert spaces under global or local Lipschitz conditions. The purpose of the present paper is to generalize the result to the case of countable time delays in Banach spaces under a global Lipschitz condition.
Keywords: Abstract evolution equations, time delays, global existence, exponential decay.
Mathematics Subject Classification: Primary: 34K30; Secondary: 34K20.
Citation: Tomomi Yokota, Kentarou Yoshii. Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation. Evolution Equations & Control Theory, doi: 10.3934/eect.2020086
References:
[1]

F. Alabau-Boussouira, S. Nicaise and C. Pignotti, Exponential stability of the wave equation with memory and time delay, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 1–22. doi: 10.1007/978-3-319-11406-4_1.  Google Scholar

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A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, AK Peters, Ltd., Wellesley, MA, 2005.  Google Scholar

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

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R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar

[7]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar

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H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.  Google Scholar

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G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Differential Equations, 5 (1993), 89-103.  doi: 10.1007/BF01063736.  Google Scholar

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A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.  Google Scholar

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A. Inoue, T. Miyakawa and K. Yoshida, Some properties of solutions for semilinear heat equations with time lag, J. Differential Equations, 24 (1977), 383-396.  doi: 10.1016/0022-0396(77)90007-9.  Google Scholar

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V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, 1994.  Google Scholar

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D. Li and S. Guo, Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect, J. Math. Anal. Appl., 467 (2018), 1080-1099.  doi: 10.1016/j.jmaa.2018.07.050.  Google Scholar

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W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56.  doi: 10.3934/dcdsb.2002.2.47.  Google Scholar

[17]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[18]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar

[19]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 2011, No. 41, 20 pp.  Google Scholar

[20]

S. Nicaise and C. Pignotti, Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.  doi: 10.1007/s00498-014-0130-1.  Google Scholar

[21]

S. Nicaise and C. Pignotti, Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15 (2015), 107-129.  doi: 10.1007/s00028-014-0251-5.  Google Scholar

[22]

S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvi–-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.  Google Scholar

[23]

S. Nicaise and C. Pignotti, Well-posedness and stability results for nonlinear abstract evolution equations with time delays, J. Evol. Equ., 18 (2018), 947-971.  doi: 10.1007/s00028-018-0427-5.  Google Scholar

[24]

S. M. Oliva, Reaction-diffusion equations with nonlinear boundary delay, J. Dynam. Differential Equations, 11 (1999), 279-296.  doi: 10.1023/A:1021929413376.  Google Scholar

[25]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  doi: 10.1006/jmaa.1996.0111.  Google Scholar

[26]

C. V. Pao, Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104.  doi: 10.1016/S1468-1218(03)00018-X.  Google Scholar

[27]

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

[28]

C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Systems Control Lett., 61 (2012), 92-97.  doi: 10.1016/j.sysconle.2011.09.016.  Google Scholar

[29]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[30]

S. Ruan and X.-Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations, 156 (1999), 71-92.  doi: 10.1006/jdeq.1998.3599.  Google Scholar

[31]

B. Said-Houari and A. Soufyane, Stability result of the Timoshenko system with delay and boundary feedback, IMA J. Math. Control Inform., 29 (2012), 383-398.  doi: 10.1093/imamci/dnr043.  Google Scholar

[32]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.  Google Scholar

[33]

K. Yoshii, Solvability in abstract evolution equations with countable time delays in Banach spaces: Lobal Lipschitz perturbation, preprint, 2020. Google Scholar

[34]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.  Google Scholar

show all references

References:
[1]

F. Alabau-Boussouira, S. Nicaise and C. Pignotti, Exponential stability of the wave equation with memory and time delay, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer INdAM Ser., Springer, Cham, 10 (2014), 1–22. doi: 10.1007/978-3-319-11406-4_1.  Google Scholar

[2]

K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems Control Lett., 59 (2010), 623-628.  doi: 10.1016/j.sysconle.2010.07.007.  Google Scholar

[3]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.  Google Scholar

[4]

A. Bátkai and S. Piazzera, Semigroups for Delay Equations, Research Notes in Mathematics, 10, AK Peters, Ltd., Wellesley, MA, 2005.  Google Scholar

[5]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.  Google Scholar

[6]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.  doi: 10.1137/0326040.  Google Scholar

[7]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.  Google Scholar

[8]

H. I. Freedman and X.-Q. Zhao, Global asymptotics in some quasimonotone reaction-diffusion systems with delays, J. Differential Equations, 137 (1997), 340-362.  doi: 10.1006/jdeq.1997.3264.  Google Scholar

[9]

G. Friesecke, Convergence to equilibrium for delay-diffusion equations with small delay, J. Dynam. Differential Equations, 5 (1993), 89-103.  doi: 10.1007/BF01063736.  Google Scholar

[10]

A. Guesmia, Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay, IMA J. Math. Control Inform., 30 (2013), 507-526.  doi: 10.1093/imamci/dns039.  Google Scholar

[11]

A. Inoue, T. Miyakawa and K. Yoshida, Some properties of solutions for semilinear heat equations with time lag, J. Differential Equations, 24 (1977), 383-396.  doi: 10.1016/0022-0396(77)90007-9.  Google Scholar

[12]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics. Masson, Paris, 1994.  Google Scholar

[13]

D. Li and S. Guo, Periodic traveling waves in a reaction-diffusion model with chemotaxis and nonlocal delay effect, J. Math. Anal. Appl., 467 (2018), 1080-1099.  doi: 10.1016/j.jmaa.2018.07.050.  Google Scholar

[14]

D. Li and S. Guo, Traveling wavefronts in a reaction-diffusion model with chemotaxis and nonlocal delay effect, Nonlinear Anal. Real World Appl., 45 (2019), 736-754.  doi: 10.1016/j.nonrwa.2018.08.001.  Google Scholar

[15]

J. H. Lightbourne and S. M. Rankin, Global existence for a delay differential equation, J. Differential Equations, 40 (1981), 186-192.  doi: 10.1016/0022-0396(81)90017-6.  Google Scholar

[16]

W. Liu, Asymptotic behavior of solutions of time-delayed Burgers' equation, Discrete Contin. Dyn. Syst. Ser. B, 2 (2002), 47-56.  doi: 10.3934/dcdsb.2002.2.47.  Google Scholar

[17]

S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.  Google Scholar

[18]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958.   Google Scholar

[19]

S. Nicaise and C. Pignotti, Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differential Equations, 2011, No. 41, 20 pp.  Google Scholar

[20]

S. Nicaise and C. Pignotti, Stabilization of second-order evolution equations with time delay, Math. Control Signals Systems, 26 (2014), 563-588.  doi: 10.1007/s00498-014-0130-1.  Google Scholar

[21]

S. Nicaise and C. Pignotti, Exponential stability of abstract evolution equations with time delay, J. Evol. Equ., 15 (2015), 107-129.  doi: 10.1007/s00028-014-0251-5.  Google Scholar

[22]

S. Nicaise and C. Pignotti, Stability of the wave equation with localized Kelvi–-Voigt damping and boundary delay feedback, Discrete Contin. Dyn. Syst. Ser. S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.  Google Scholar

[23]

S. Nicaise and C. Pignotti, Well-posedness and stability results for nonlinear abstract evolution equations with time delays, J. Evol. Equ., 18 (2018), 947-971.  doi: 10.1007/s00028-018-0427-5.  Google Scholar

[24]

S. M. Oliva, Reaction-diffusion equations with nonlinear boundary delay, J. Dynam. Differential Equations, 11 (1999), 279-296.  doi: 10.1023/A:1021929413376.  Google Scholar

[25]

C. V. Pao, Dynamics of nonlinear parabolic systems with time delays, J. Math. Anal. Appl., 198 (1996), 751-779.  doi: 10.1006/jmaa.1996.0111.  Google Scholar

[26]

C. V. Pao, Global asymptotic stability of Lotka–Volterra competition systems with diffusion and time delays, Nonlinear Anal. Real World Appl., 5 (2004), 91-104.  doi: 10.1016/S1468-1218(03)00018-X.  Google Scholar

[27]

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

[28]

C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Systems Control Lett., 61 (2012), 92-97.  doi: 10.1016/j.sysconle.2011.09.016.  Google Scholar

[29]

P. Quittner and P. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2007.  Google Scholar

[30]

S. Ruan and X.-Q. Zhao, Persistence and extinction in two species reaction-diffusion systems with delays, J. Differential Equations, 156 (1999), 71-92.  doi: 10.1006/jdeq.1998.3599.  Google Scholar

[31]

B. Said-Houari and A. Soufyane, Stability result of the Timoshenko system with delay and boundary feedback, IMA J. Math. Control Inform., 29 (2012), 383-398.  doi: 10.1093/imamci/dnr043.  Google Scholar

[32]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.  Google Scholar

[33]

K. Yoshii, Solvability in abstract evolution equations with countable time delays in Banach spaces: Lobal Lipschitz perturbation, preprint, 2020. Google Scholar

[34]

E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.  doi: 10.1080/03605309908820684.  Google Scholar

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