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.
References
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. MR2592579 1193.34149 10.1016/j.apm.2009.08.027 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. MR2592579 1193.34149 10.1016/j.apm.2009.08.027
T. Caraballo, Nonlinear partial differential equations: Existence and stability, J. Math. Anal. Appl., 262 (2001), 87-111. 0991.35098 10.1006/jmaa.2001.7545 T. Caraballo, Nonlinear partial differential equations: Existence and stability, J. Math. Anal. Appl., 262 (2001), 87-111. 0991.35098 10.1006/jmaa.2001.7545
T. Caraballo and J. Real, Navier-Stokes equations with delays, Proc. R. Soc. London Ser., A 457 (2001), 2441-2454. 1007.35062 10.1098/rspa.2001.0807 T. Caraballo and J. Real, Navier-Stokes equations with delays, Proc. R. Soc. London Ser., A 457 (2001), 2441-2454. 1007.35062 10.1098/rspa.2001.0807
T. Caraballo and J. Real, Asymptotic behaviour of Navier-Stokes equations with delays, Proc. R. Soc. London Ser., A 459 (2003), 3181-3194. 1057.35027 10.1098/rspa.2003.1166 T. Caraballo and J. Real, Asymptotic behaviour of Navier-Stokes equations with delays, Proc. R. Soc. London Ser., A 459 (2003), 3181-3194. 1057.35027 10.1098/rspa.2003.1166
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. 1178.34087 10.1016/j.jmaa.2007.01.038 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. 1178.34087 10.1016/j.jmaa.2007.01.038
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. 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. 1100.37047 I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Acta, 2002. 1100.37047
M. J. Garrido-Atienza and P. Mar\iprefff n-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. 1091.35058 10.1016/j.na.2005.05.057 M. J. Garrido-Atienza and P. Mar\iprefff n-Rubio, Navier-Stokes equations with delays on unbounded domains, Nonlinear Anal., 64 (2006), 1100-1118. 1091.35058 10.1016/j.na.2005.05.057
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, 1993. 0787.34002 J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer, 1993. 0787.34002
E. Fridman and Y. Orlov, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45 (2009), 194-201. 1154.93404 10.1016/j.automatica.2008.06.006 E. Fridman and Y. Orlov, Exponential stability of linear distributed parameter systems with time-varying delays, Automatica, 45 (2009), 194-201. 1154.93404 10.1016/j.automatica.2008.06.006
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. 1180.35095 10.1137/060648891 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. 1180.35095 10.1137/060648891
V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York, 1989. 0676.34003 V. Lakshmikantham, S. Leela and A. A. Martynyuk, Stability Analysis of Nonlinear Systems, Marcel Dekker, Inc., New York, 1989. 0676.34003
J. So and J. S. Yu, Global attractivity and uniform persistence in Nichonson's blowflies, Diff. Equa. Dynam. Systems, 2 (1994), 11-18. 0869.34056 J. So and J. S. Yu, Global attractivity and uniform persistence in Nichonson's blowflies, Diff. Equa. Dynam. Systems, 2 (1994), 11-18. 0869.34056
T. Wang, Exponent stability and inequalities of solutions of abstract functional differential equations, J. Math. Anal. Appl., 324 (2006), 982-991. 1112.34060 10.1016/j.jmaa.2005.12.076 T. Wang, Exponent stability and inequalities of solutions of abstract functional differential equations, J. Math. Anal. Appl., 324 (2006), 982-991. 1112.34060 10.1016/j.jmaa.2005.12.076
J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996. 0870.35116 J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, 1996. 0870.35116