# American Institute of Mathematical Sciences

May  2015, 20(3): 781-810. doi: 10.3934/dcdsb.2015.20.781

## Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces

 1 Department of Mathematics, University of Surrey, Guildford, GU2 7XH

Received  April 2014 Revised  August 2014 Published  January 2015

We give a comprehensive study of strong uniform attractors of nonautonomous dissipative systems for the case where the external forces are not translation compact. We introduce several new classes of external forces that are not translation compact, but nevertheless allow the attraction in a strong topology of the phase space to be verified and discuss in a more detailed way the class of so-called normal external forces introduced before. We also develop a unified approach to verify the asymptotic compactness for such systems based on the energy method and apply it to a number of equations of mathematical physics including the Navier-Stokes equations, damped wave equations and reaction-diffusing equations in unbounded domains.
Citation: Sergey Zelik. Strong uniform attractors for non-autonomous dissipative PDEs with non translation-compact external forces. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 781-810. doi: 10.3934/dcdsb.2015.20.781
##### References:
 [1] C. Anh and N. Quang, Uniform attractors for nonautonomous parabolic equations involving weighted p-Laplacian operators, Ann. Polon. Math., 98 (2010), 251-271. doi: 10.4064/ap98-3-5. [2] A. Babin and M. Vishik, Attractors of Evolutionary Equations, North Holland, Amsterdam, 1992. [3] J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [4] A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [5] V. Chepyzhov, On uniform attractors of dynamic processes and nonautonomous equations of mathematical physics, Russian Math. Surveys, 68 (2013), 349-382. [6] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [7] V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. [8] V. Chepyzhov and M. Vishik, Attractors of non-autonomous evolution equations with translation-compact symbols, in Partial Differential Operators and Mathematical Physics (Holzhau, 1994), Oper. Theory Adv. Appl., 78, Birkhäuser, Basel, 1995, 49-60. [9] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [10] M. Efendiev, S. Zelik and A. Miranville, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X. [11] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, submitted. [12] S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. [13] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: 10.1016/j.jde.2006.07.009. [14] S. Lu., Attractors for nonautonomous reaction-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210. [15] S. Ma, C. Zhong and H. Song, Attractors for nonautonomous 2D Navier-Stokes equations with less regular symbols, Nonlinear Anal., 71 (2009), 4215-4222. doi: 10.1016/j.na.2009.02.107. [16] S. Ma, X. Cheng and H. Li, Attractors for non-autonomous wave equations with a new class of external forces, J. Math. Anal. Appl., 337 (2008), 808-820. doi: 10.1016/j.jmaa.2007.03.108. [17] S. Ma and C. Zhong, The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dyn. Syst., 18 (2007), 53-70. doi: 10.3934/dcds.2007.18.53. [18] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0. [19] I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393. doi: 10.1088/0951-7715/11/5/012. [20] I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473. [21] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [22] A. Robertson and W. Robertson, Topological Vector Spaces, Reprint of the second edition, Cambridge Tracts in Mathematics, 53, Cambridge University Press, Cambridge-New York, 1980. [23] Y. Xie, K. Zhu and C. Sun, The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space, J. Math. Phys., 53 (2012), 082703, 11 pp. doi: 10.1063/1.4746693.

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##### References:
 [1] C. Anh and N. Quang, Uniform attractors for nonautonomous parabolic equations involving weighted p-Laplacian operators, Ann. Polon. Math., 98 (2010), 251-271. doi: 10.4064/ap98-3-5. [2] A. Babin and M. Vishik, Attractors of Evolutionary Equations, North Holland, Amsterdam, 1992. [3] J. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst., 10 (2004), 31-52. doi: 10.3934/dcds.2004.10.31. [4] A. Carvalho, J. Langa and J. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182, Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4. [5] V. Chepyzhov, On uniform attractors of dynamic processes and nonautonomous equations of mathematical physics, Russian Math. Surveys, 68 (2013), 349-382. [6] V. Chepyzhov and M. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49, American Mathematical Society, Providence, RI, 2002. [7] V. Chepyzhov and M. Vishik, Attractors of nonautonomous dynamical systems and their dimension, J. Math. Pures Appl., 73 (1994), 279-333. [8] V. Chepyzhov and M. Vishik, Attractors of non-autonomous evolution equations with translation-compact symbols, in Partial Differential Operators and Mathematical Physics (Holzhau, 1994), Oper. Theory Adv. Appl., 78, Birkhäuser, Basel, 1995, 49-60. [9] H. Crauel and F. Flandoli, Attractors for random dynamical systems, Probab. Theory Related Fields, 100 (1994), 365-393. doi: 10.1007/BF01193705. [10] M. Efendiev, S. Zelik and A. Miranville, Exponential attractors and finite-dimensional reduction for non-autonomous dynamical systems, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 703-730. doi: 10.1017/S030821050000408X. [11] V. Kalantarov, A. Savostianov and S. Zelik, Attractors for damped quintic wave equations in bounded domains, submitted. [12] S. Lu, H. Wu and C. Zhong, Attractors for nonautonomous 2D Navier-Stokes equations with normal external forces, Discrete Contin. Dyn. Syst., 13 (2005), 701-719. doi: 10.3934/dcds.2005.13.701. [13] S. Lu, Attractors for nonautonomous 2D Navier-Stokes equations with less regular normal forces, J. Differential Equations, 230 (2006), 196-212. doi: 10.1016/j.jde.2006.07.009. [14] S. Lu., Attractors for nonautonomous reaction-diffusion systems with symbols without strong translation compactness, Asymptot. Anal., 54 (2007), 197-210. [15] S. Ma, C. Zhong and H. Song, Attractors for nonautonomous 2D Navier-Stokes equations with less regular symbols, Nonlinear Anal., 71 (2009), 4215-4222. doi: 10.1016/j.na.2009.02.107. [16] S. Ma, X. Cheng and H. Li, Attractors for non-autonomous wave equations with a new class of external forces, J. Math. Anal. Appl., 337 (2008), 808-820. doi: 10.1016/j.jmaa.2007.03.108. [17] S. Ma and C. Zhong, The attractors for weakly damped non-autonomous hyperbolic equations with a new class of external forces, Discrete Contin. Dyn. Syst., 18 (2007), 53-70. doi: 10.3934/dcds.2007.18.53. [18] A. Miranville and S. Zelik, Attractors for dissipative partial differential equations in bounded and unbounded domains, in Handbook of Differential Equations: Evolutionary Equations. Vol. IV, Handb. Differ. Equ., Elsevier/North-Holland, Amsterdam, 2008, 103-200. doi: 10.1016/S1874-5717(08)00003-0. [19] I. Moise, R. Rosa and X. Wang, Attractors for non-compact semigroups via energy equations, Nonlinearity, 11 (1998), 1369-1393. doi: 10.1088/0951-7715/11/5/012. [20] I. Moise, R. Rosa and X. Wang, Attractors for noncompact nonautonomous systems via energy equations, Discrete Contin. Dyn. Syst., 10 (2004), 473-496. doi: 10.3934/dcds.2004.10.473. [21] R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics, Second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. doi: 10.1007/978-1-4612-0645-3. [22] A. Robertson and W. Robertson, Topological Vector Spaces, Reprint of the second edition, Cambridge Tracts in Mathematics, 53, Cambridge University Press, Cambridge-New York, 1980. [23] Y. Xie, K. Zhu and C. Sun, The existence of uniform attractors for non-autonomous reaction-diffusion equations on the whole space, J. Math. Phys., 53 (2012), 082703, 11 pp. doi: 10.1063/1.4746693.
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