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

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A. N. CarvalhoJ. A. LangaJ. C. Robinson and A. Suárez, Characterization of nonautonomous attractors of a perturbed gradient system, J. Differential Equations, 236 (2007), 570-603.   Google Scholar

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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

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T. CaraballoP. 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. CaraballoJ. 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

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

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D. ChebanP. 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-CostaT. CaraballoA. N. Carvalho and J. A. Langa, Stability of gradient semigroups under perturbation, Nonlinearity, 24 (2011), 2099-2117.   Google Scholar

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

[13]

H. CrauelA. 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

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

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J. DuanK. 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. VishikS. V. Zelik and V. V. Chepyzhov, Regular attractors and nonautonomous perturbations of them, Mat. Sb., 204 (2013), 1-42.   Google Scholar

show all references

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. CarvalhoJ. A. LangaJ. 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. CaraballoP. 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. CaraballoJ. 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. ChebanP. 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-CostaT. CaraballoA. 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. CrauelA. 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. DuanK. Lu and B. Schmalfuss, Invariant manifolds for stochastic partial differential equations, Ann. Probab., 31 (2003), 2109-2135.   Google Scholar

[16]

J. DuanK. 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. VishikS. 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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