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

Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing

Xuewei Ju 1,, and  Desheng Li 2,

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.
Keywords: Nonautonomous dynamical systems, nonautonomous hyperbolic equilibrium, invariant manifolds of equilibrium, stochastic hyperbolic equilibrium, global synchronising behavior.
Mathematics Subject Classification: 37B55, 37D05, 37L55, 34D06, 37D10.
Citation: Xuewei Ju, Desheng Li. Global synchronising behavior of evolution equations with exponentially growing nonautonomous forcing. Communications on Pure and Applied Analysis, 2018, 17 (5) : 1921-1944. doi: 10.3934/cpaa.2018091
References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998.

[2]

P. Brune and B. Schmalfuss, Inertial manifolds for stochastic pde with dynamical boundary conditions, Commun. Pure Appl. Anal., 10 (2011), 831-846. 

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci. 182, 2013.

[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. 

[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. 

[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. 

[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. 

[8]

X. Chen and J. Duan, State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974. 

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.

[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. 

[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. 

[12]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. 

[13]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. 

[14]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.

[15]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. 

[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. 

[17]

J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R. I., 1989.

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981.

[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. 

[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. 

[22]

D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1.

[23]

K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. 

[24]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.

[25]

M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42. 

show all references

References:
[1]

L. Arnold, Random Dynamical Systems, Springer, New York, 1998.

[2]

P. Brune and B. Schmalfuss, Inertial manifolds for stochastic pde with dynamical boundary conditions, Commun. Pure Appl. Anal., 10 (2011), 831-846. 

[3]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Appl. Math. Sci. 182, 2013.

[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. 

[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. 

[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. 

[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. 

[8]

X. Chen and J. Duan, State space decomposition for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 141 (2011), 957-974. 

[9]

V. V. Chepyzhov and M. I. Vishik, Attractors of Equations of Mathematical Physics, Amer. Math. Soc., Providence, RI, 2002.

[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. 

[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. 

[12]

H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. 

[13]

H. Crauel, A. Debussche and F. Flandoli, Random attractors, J. Dynam. Differential Equations, 9 (1997), 307-341. 

[14]

G. Da Prato and J. Zabczyk, Stochastic Equations in Infinite Dimensions, Cambridge Univ. Press, Cambridge, 1992.

[15]

J. Duan, K. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135. 

[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. 

[17]

J. Duan and W. Wang, Effective Dynamics of Stochastic Partial Differential Equations, Elsevier, 2014.

[18]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr. 25, Amer. Math. Soc., R. I., 1989.

[19]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math. 840, Springer-Verlag, 1981.

[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. 

[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. 

[22]

D. S. Li, Morse theory of attractors via Lyapunov functions, preprint, arXiv: 1003.0305v1.

[23]

K. Lu and B. Schmalfuss, Invariant manifolds for stochastic wave equations, J. Differential Equations, 236 (2007), 460-492. 

[24]

J. C. Robinson, Infinite-Dimensional Dynamical Systems, Cambridge Univ. Press, Cambridge, 2001.

[25]

M. I. Vishik, S. V. Zelik and V. V. Chepyzhov, Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42. 

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