Abstract
We consider abstract evolution equations with a nonlinear term depending on the state and on delayed states. We show that, if the \(C_0\)-semigroup describing the linear part of the model is exponentially stable, then the whole system retains this property under some Lipschitz continuity assumptions on the nonlinearity. More precisely, we give a general exponential decay estimate for small time delays if the nonlinear term is globally Lipschitz and an exponential decay estimate for solutions starting from small data when the nonlinearity is only locally Lipschitz and the linear part is a negative selfadjoint operator. In the latter case we do not need any restriction on the size of the time delays. In both cases, concrete examples are presented that illustrate our abstract results.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
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., 10:1–22, 2014.
K. Ammari, S. Nicaise and C. Pignotti. Feedback boundary stabilization of wave equations with interior delay. Systems and Control Lett., 59:623–628, 2010.
A. Bátkai and S. Piazzera. Semigroups for delay equations, Research Notes in Mathematics, 10. AK Peters, Ltd., Wellesley, MA, 2005.
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., 30:1024–1065, 1992.
C. M. Dafermos. Asymptotic stability in viscoelasticity. Arch. Rational Mech. Anal., 37:297–308, 1970.
R. Datko. Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks. SIAM J. Control Optim., 26:697–713, 1988.
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., 24:152–156, 1986.
H. I. Freedman and X. Q. Zhao. Global asymptotics in some quasimonotone reaction-diffusion systems with delays. J. Differential Equations, 137(2):340–362, 1997.
G. Friesecke. Convergence to equilibrium for delay-diffusion equations with small delay. J. Dynam. Differential Equations, 5:89–103, 1993.
C. Giorgi, J. E. Muñoz Rivera and V. Pata. Global attractors for a semilinear hyperbolic equation in viscoelasticity. J. Math. Anal. Appl., 260:83–99, 2001.
P. Grisvard. Elliptic problems in nonsmooth domains, volume 24 of Monographs and Studies in Mathematics. Pitman, Boston–London–Melbourne, 1985.
A. Guesmia. Well-posedness and exponential stability of an abstract evolution equation with infinite memory and time delay. IMA J. Math. Control Inform., 30:507–526, 2013.
A. Inoue, T. Miyakawa and K. Yoshida. Some properties of solutions for semilinear heat equations with time lag. J. Differential Equations, 24(3):383–396, 1977.
T. Kato. Perturbation theory for linear operators. Springer-Verlag, Berlin, 1976.
V. Komornik. Exact controllability and stabilization, the multiplier method, volume 36 of RMA. Masson, Paris, 1994.
J. H. Lightbourne and S. M. Rankin. Global existence for a delay differential equation. J. Differential Equations, 40(2):186–192, 1981.
W. Liu. Asymptotic behavior of solutions of time-delayed Burgers’ equation. Discrete Contin. Dyn. Syst. Ser. B, 2(1):47–56, 2002.
M. C. Memory. Stable and unstable manifolds for partial functional-differential equations. Nonlinear Anal., 16(2):131–142, 1991.
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., 45:1561–1585, 2006.
S. Nicaise and C. Pignotti. Stabilization of second-order evolution equations with time delay. Math. Control Signals Systems, 26:563–588, 2014.
S. Nicaise and C. Pignotti. Exponential stability of abstract evolution equations with time delay. J. Evol. Equ., 15:107–129, 2015.
S. M. Oliva. Reaction-diffusion equations with nonlinear boundary delay. J. Dynam. Differential Equations, 11:279–296, 1999.
C. V. Pao. Dynamics of nonlinear parabolic systems with time delays. J. Math. Anal. Appl., 198(3):751–779, 1996.
C. V. Pao. Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays. Nonlinear Anal. Real World Appl., 5(1):91–104, 2004.
A. Pazy. Semigroups of linear operators and applications to partial differential equations, Vol. 44 of Applied Math. Sciences. Springer-Verlag, New York, 1983.
C. Pignotti. A note on stabilization of locally damped wave equations with time delay. Systems and Control Lett., 61:92–97, 2012.
S. Ruan and X. Q. Zhao. Persistence and extinction in two species reaction-diffusion systems with delays. J. Differential Equations, 156(1):71–92, 1999.
B. Said-Houari and A. Soufyane. Stability result of the Timoshenko system with delay and boundary feedback. IMA J. Math. Control Inform., 29:383–398, 2012.
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(4):770–785, 2006.
E. Zuazua. Exponential decay for the semi-linear wave equation with locally distributed damping. Comm. Partial Differential Equations, 15:205–235, 1990.
Acknowledgements
The research of the second author is partially supported by the GNAMPA 2016 project Controllo, regolarità e viabilità per alcuni tipi di equazioni diffusive and the GNAMPA 2017 project Comportamento asintotico e controllo di equazioni di evoluzione non lineari (INdAM).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Nicaise, S., Pignotti, C. Well-posedness and stability results for nonlinear abstract evolution equations with time delays. J. Evol. Equ. 18, 947–971 (2018). https://doi.org/10.1007/s00028-018-0427-5
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00028-018-0427-5