June  2022, 15(6): 1561-1571. doi: 10.3934/dcdss.2022049

Well-posedness and stability for semilinear wave-type equations with time delay

Dipartimento di Ingegneria e Scienze dell'Informazione e Matematica, Università dell'Aquila, Via Vetoio, 67010, L'Aquila, Italy

* Corresponding author: Cristina Pignotti

Received  August 2021 Revised  January 2022 Published  June 2022 Early access  March 2022

In this paper, we analyze a semilinear abstract damped wave-type equation with time delay. We assume that the delay feedback coefficient is variable in time and belongs to $ L^1_{loc}([0, +\infty)). $ Under suitable assumptions, we show well-posedness and exponential stability for small initial data. Our strategy combines careful energy estimates and continuity arguments. Some examples illustrate the abstract results.

Citation: Alessandro Paolucci, Cristina Pignotti. Well-posedness and stability for semilinear wave-type equations with time delay. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1561-1571. doi: 10.3934/dcdss.2022049
References:
[1]

E. M. Ait BenhassiK. AmmariS. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay, J. Evol. Equ., 9 (2009), 103-121.  doi: 10.1007/s00028-009-0004-z.

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[3]

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

[4]

K. Ammari and B. Chentouf, Asymptotic behavior of a delayed wave equation without displacement term, Z. Angew. Math. Phys., 68 (2017), Paper No. 117, 13 pp. doi: 10.1007/s00033-017-0865-x.

[5]

K. Ammari and S. Gerbi, Interior feedback stabilization of wave equations with dynamic boundary delay, Z. Anal. Anwend., 36 (2017), 297-327.  doi: 10.4171/ZAA/1590.

[6]

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.

[7]

T. A. Apalara and S. A. Messaoudi, An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay, Appl. Math. Optim., 71 (2015), 449-472.  doi: 10.1007/s00245-014-9266-0.

[8]

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.

[9]

Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6.

[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]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris and John Wiley & Sons, Chicester, 1994.

[12]

V. Komornik and C. Pignotti, Energy decay for evolution equations with delay feedbacks, Math. Nachr., 295 (2022), 377-394.  doi: 10.1002/mana.201900532.

[13]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[14]

M. I. Mustafa and M. Kafini, Energy decay for viscoelastic plates with distributed delay and source term, Z. Angew. Math. Phys., 67 (2016), Paper No. 36, 18 pp. doi: 10.1007/s00033-016-0641-3.

[15]

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.

[16]

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.

[17]

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.

[18]

R. L. Oliveira and H. P. Oquendo, Stability and instability results for coupled waves with delay term, J. Math. Phys., 61 (2020), 071505, 13 pp. doi: 10.1063/1.5144987.

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Math. Sciences. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[20]

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

[21]

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.

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts, Birkhäuser-Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[23]

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.

show all references

References:
[1]

E. M. Ait BenhassiK. AmmariS. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay, J. Evol. Equ., 9 (2009), 103-121.  doi: 10.1007/s00028-009-0004-z.

[2]

F. Alabau-BoussouiraP. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[3]

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

[4]

K. Ammari and B. Chentouf, Asymptotic behavior of a delayed wave equation without displacement term, Z. Angew. Math. Phys., 68 (2017), Paper No. 117, 13 pp. doi: 10.1007/s00033-017-0865-x.

[5]

K. Ammari and S. Gerbi, Interior feedback stabilization of wave equations with dynamic boundary delay, Z. Anal. Anwend., 36 (2017), 297-327.  doi: 10.4171/ZAA/1590.

[6]

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.

[7]

T. A. Apalara and S. A. Messaoudi, An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay, Appl. Math. Optim., 71 (2015), 449-472.  doi: 10.1007/s00245-014-9266-0.

[8]

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.

[9]

Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6.

[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]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Masson, Paris and John Wiley & Sons, Chicester, 1994.

[12]

V. Komornik and C. Pignotti, Energy decay for evolution equations with delay feedbacks, Math. Nachr., 295 (2022), 377-394.  doi: 10.1002/mana.201900532.

[13]

J.-L. Lions, Exact controllability, stabilizability, and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[14]

M. I. Mustafa and M. Kafini, Energy decay for viscoelastic plates with distributed delay and source term, Z. Angew. Math. Phys., 67 (2016), Paper No. 36, 18 pp. doi: 10.1007/s00033-016-0641-3.

[15]

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.

[16]

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.

[17]

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.

[18]

R. L. Oliveira and H. P. Oquendo, Stability and instability results for coupled waves with delay term, J. Math. Phys., 61 (2020), 071505, 13 pp. doi: 10.1063/1.5144987.

[19]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Vol. 44 of Applied Math. Sciences. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[20]

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

[21]

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.

[22]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts, Birkhäuser-Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

[23]

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.

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