American Institute of Mathematical Sciences

September  2018, 17(5): 1921-1944. doi: 10.3934/cpaa.2018091

Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing

 1 Department of Mathematics, School of Science, Civil Aviation University of China, Tianjin 300300, China 2 School of Mathematics, Tianjin University, Tianjin 300072, China

* Corresponding author

Received  August 2017 Revised  December 2017 Published  April 2018

This work is concerned with the following nonautonomous evolutionary system on a Banach space
 $X$
,
 ${x_t} + Ax = f\left( {x, h\left( t \right)} \right), \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {0.1} \right)$
where
 $A$
is a hyperbolic sectorial operator on
 $X$
, the nonlinearity
 $f \in C({X^\alpha } \times X,X)$
is Lipschitz in the first variable, the nonautonomous forcing
 $h \in C(\mathbb{R},X)$
is
 $\mu$
-subexponentially growing for some
 $\mu >0$
(see (3.4) below for definition). Under some reasonable assumptions, we first establish an existence result for a unique nonautonomous hyperbolic equilibrium for the system in the framework of cocycle semiflows. We then demonstrate that the system exhibits a global synchronising behavior with the nonautonomous forcing
 $h$
as time varies. Finally, we apply the abstract results to stochastic partial differential equations with additive white noise and obtain stochastic hyperbolic equilibria for the corresponding systems.
Citation: Xuewei Ju, Desheng Li. Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing. Communications on Pure & Applied Analysis, 2018, 17 (5) : 1921-1944. doi: 10.3934/cpaa.2018091
References:
 [1] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. Google Scholar [2] P. Brune and B. Schmalfuss, Inertial manifolds for stochastic pde with dynamical boundary conditions, Commun. Pure Appl. Anal., 10 (2011), 831-846.   Google Scholar [3] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci. 182, 2013. Google Scholar [4] A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.   Google Scholar [5] A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.   Google Scholar [6] T. Caraballo, P. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.   Google Scholar [7] T. Caraballo, J. A. Langa and Z. X. Liu, Gradient infinite-dimensional random dynamical system, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.   Google Scholar [8] X. Chen and J. Duan, State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974.   Google Scholar [9] V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. Google Scholar [10] D. Cheban, P. Kloeden and B. Schmalfuss, The relation between pullback and global attractors for nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.   Google Scholar [11] E. R. Arag$\tilde{a}$o-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.   Google Scholar [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.   Google Scholar [13] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.   Google Scholar [14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. Google Scholar [15] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.   Google Scholar [16] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.   Google Scholar [17] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014. Google Scholar [18] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R. I., 1989. Google Scholar [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981. Google Scholar [20] D. S. Li and X. X. Zhang, On dynamical properties of general dynamical systems and differential inclusions, J. Appl. Math. Anal. Appl., 274 (2002), 705-724.   Google Scholar [21] D. S. Li and J. Duan, Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations, J. Differential Equations, 246 (2009), 1754-1773.   Google Scholar [22] D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. Google Scholar [23] K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.   Google Scholar [24] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. Google Scholar [25] M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42.   Google Scholar

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References:
 [1] L. Arnold, Random Dynamical Systems, Springer, New York, 1998. Google Scholar [2] P. Brune and B. Schmalfuss, Inertial manifolds for stochastic pde with dynamical boundary conditions, Commun. Pure Appl. Anal., 10 (2011), 831-846.   Google Scholar [3] A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci. 182, 2013. Google Scholar [4] A. N. Carvalho, J. A. Langa, J. C. Robinson and A. Suárez, Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.   Google Scholar [5] A. N. Carvalho and J. A. Langa, An extension of the concept of gradient semigroups which is stable under perturbation, J. Differential Equations, 246 (2009), 2646-2668.   Google Scholar [6] T. Caraballo, P. Kloeden and B. Schmalfuss, Exponentially stable stationary solutions for stochastic evolution equations and their perturbation, Appl. Math. Optim., 50 (2004), 183-207.   Google Scholar [7] T. Caraballo, J. A. Langa and Z. X. Liu, Gradient infinite-dimensional random dynamical system, SIAM J. Appl. Dyn. Syst., 11 (2012), 1817-1847.   Google Scholar [8] X. Chen and J. Duan, State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974.   Google Scholar [9] V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002. Google Scholar [10] D. Cheban, P. Kloeden and B. Schmalfuss, The relation between pullback and global attractors for nonautonomous dynamical systems, Nonlinear Dyn. Syst. Theory, 2 (2002), 125-144.   Google Scholar [11] E. R. Arag$\tilde{a}$o-Costa, T. Caraballo, A. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.   Google Scholar [12] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393.   Google Scholar [13] H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341.   Google Scholar [14] G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992. Google Scholar [15] J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.   Google Scholar [16] J. Duan, K. Lu and B. Schmalfuss, Smooth stable and unstable manifolds for stochastic evolutionary equations, J. Dynam. Differential Equations, 16 (2004), 949-972.   Google Scholar [17] J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014. Google Scholar [18] J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R. I., 1989. Google Scholar [19] D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981. Google Scholar [20] D. S. Li and X. X. Zhang, On dynamical properties of general dynamical systems and differential inclusions, J. Appl. Math. Anal. Appl., 274 (2002), 705-724.   Google Scholar [21] D. S. Li and J. Duan, Structure of the set of bounded solutions for a class of nonautonomous second-order differential equations, J. Differential Equations, 246 (2009), 1754-1773.   Google Scholar [22] D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1. Google Scholar [23] K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492.   Google Scholar [24] J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001. Google Scholar [25] M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42.   Google Scholar
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