Taiwanese Journal of Mathematics


Cung The Anh and Le Van Hien

Full-text: Open access


In this paper we prove the global existence and the global exponential stability of weak solutions to a class of semilinear parabolic equations with discrete and distributed time-varying delays. Moreover, the exponential stability of stationary solutions of the equations is also studied. The obtained results can be applied to some models in biology and physics.

Article information

Taiwanese J. Math., Volume 16, Number 6 (2012), 2133-2151.

First available in Project Euclid: 18 July 2017

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35B35: Stability 37L15: Stability problems

PDEs with delays weak solution stationary solution exponential stability Galerkin method Lyapunov-Krasovskii functional


Anh, Cung The; Hien, Le Van. EXPONENTIAL STABILITY OF SOLUTIONS TO SEMILINEAR PARABOLIC EQUATIONS WITH DELAYS. Taiwanese J. Math. 16 (2012), no. 6, 2133--2151. doi:10.11650/twjm/1500406844. https://projecteuclid.org/euclid.twjm/1500406844

Export citation


  • L. Berezansky, E. Braverman and L. Idels, Nicholson's blowflies differential equations revisited: Main results and open problems, Appl. Math. Modelling, 34 (2010), 1405-1417.
  • T. Caraballo, Nonlinear partial differential equations: Existence and stability, J. Math. Anal. Appl., 262 (2001), 87-111.
  • T. Caraballo and J. Real, Navier-Stokes equations with delays, Proc. R. Soc. London Ser., A 457 (2001), 2441-2454.
  • T. Caraballo and J. Real, Asymptotic behaviour of Navier-Stokes equations with delays, Proc. R. Soc. London Ser., A 459 (2003), 3181-3194.
  • T. Caraballo, J. Real and L. Shaikhet, Method of Lyapunov functionals construction in stability of delay evolution equations, J. Math. Anal. Appl., 334 (2007), 1130-1145.
  • T. Caraballo, A. M. Márquez-Durán and J. Real, Asymptotic behaviour of the three-dimensional $\alpha$-Navier-Stokes model with delays, J. Math. Anal. Appl., 340 (2008), 410-423.
  • I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, 2002.
  • M. J. Garrido-Atienza and P. Mar\iprefff n-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118.
  • J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, 1993.
  • E. Fridman and Y. Orlov, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45 (2009), 194-201.
  • S. Nicaise and 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.
  • V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York, 1989.
  • J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.
  • J. So and J. S. Yu, Global attractivity and uniform persistence in Nichonson's blowflies, Diff. Equa. Dynam. Systems, 2 (1994), 11-18.
  • T. Wang, Exponent stability and inequalities of solutions of abstract functional differential equations, J. Math. Anal. Appl., 324 (2006), 982-991.
  • J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996.