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December  2021, 10(4): 689-699. doi: 10.3934/eect.2020086

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

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 Published  December 2021 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.
Citation: Tomomi Yokota, Kentarou Yoshii. Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation. Evolution Equations and Control Theory, 2021, 10 (4) : 689-699. 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.

[2]

K. AmmariS. 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.

[3]

C. BardosG. 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.

[4]

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

[5]

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

[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.

[7]

R. DatkoJ. 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.

[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.

[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.

[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.

[11]

A. InoueT. 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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

G. Q. XuS. 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.

[33]

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

[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.

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.

[2]

K. AmmariS. 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.

[3]

C. BardosG. 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.

[4]

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

[5]

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

[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.

[7]

R. DatkoJ. 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.

[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.

[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.

[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.

[11]

A. InoueT. 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.

[12]

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

[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.

[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.

[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.

[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.

[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.

[18]

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

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[32]

G. Q. XuS. 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.

[33]

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

[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.

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