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March  2022, 21(3): 943-958. doi: 10.3934/cpaa.2022005

## Inertial manifolds for parabolic differential equations: The fully nonautonomous case

 1 School of Applied Mathematics and Informatics, Hanoi University of Science and Technology, Vien Toan ung dung va Tin hoc, Dai hoc Bach khoa Hanoi, 1 Dai Co Viet, Hanoi, Vietnam 2 Faculty of Computer Science and Engineering, Department of Mathematics, Thuyloi University, Khoa Cong nghe Thong tin, Bo mon Toan, Dai hoc Thuy loi, 175 Tay Son, Dong Da, Ha Noi, Vietnam 3 Fermat Education, 6A1-Ngoc Khanh, Ba Dinh, Hanoi, Vietnam

*Corresponding author

Received  July 2021 Revised  November 2021 Published  March 2022 Early access  December 2021

Fund Project: The work of V.T.N. Ha is partly supported by by the Project of Vietnam Ministry of Education and Training under Project B2022-BKA-06. This work is financially supported by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant No. 101.02-2021.04

We study the existence of an inertial manifold for the solutions to fully non-autonomous parabolic differential equation of the form
 $\dfrac{du}{dt} + A(t)u(t) = f(t,u),\, t> s.$
We prove the existence of such an inertial manifold in the case that the family of linear partial differential operators
 $(A(t))_{t\in { \mathbb {R}}}$
generates an evolution family
 $(U(t,s))_{t\ge s}$
satisfying certain dichotomy estimates, and the nonlinear forcing term
 $f(t,x)$
satisfies the Lipschitz condition such that certain dichotomy gap condition holds.
Citation: Nguyen Thieu Huy, Pham Truong Xuan, Vu Thi Ngoc Ha, Vu Thi Thuy Ha. Inertial manifolds for parabolic differential equations: The fully nonautonomous case. Communications on Pure and Applied Analysis, 2022, 21 (3) : 943-958. doi: 10.3934/cpaa.2022005
##### References:
 [1] P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107. [2] P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differ. Integral Equ., 1 (1988), 433-457. [3] I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differ. Equ., 13 (2001), 355-380.  doi: 10.1023/A:1016684108862. [4] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer, 1989. doi: 10.1007/BF01048790. [5] A. Debussche and R. Temam, Inertial manifolds and the slow manifolds in meteorology, Differ. Integral Equ., 4 (1991), 897-931. [6] A. Debussche and R. Temam, Some new generalizations of inertial manifolds, Discrete and Continuous Dynamical Systems, 2 (1996), 543-558.  doi: 10.3934/dcds.1996.2.543. [7] C. Foias, G. R. Sell, and R. Temam, Variétés inertielles des équations différentielles dissipatives, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 301 (1985), 139–142. [8] K. Furuya and A. Yagi, Linearized stability for abstract quasilinear equations of parabolic type, Funkcial. Ekvac., 37 (1994), 483-504. [9] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer, 2000. [10] N. T. Huy, Inertial Manifolds for Semi-linear Parabolic Equations in Admissible Spaces, J. Math. Anal. Appl., 386 (2012), 894-909.  doi: 10.1016/j.jmaa.2011.08.051. [11] N. T. Huy, Admissibly inertial manifolds for a class of semi-linear evolution equations, J. Differ. Equ., 254 (2013), 2638-2660.  doi: 10.4064/ap112-2-3. [12] N. T. Huy and X. Q. Bui, Competition models with diffusion, analytic semigroups, and inertial manifolds, Math. Method. Appl. Sci., 41 (2018), 8182-8200.  doi: 10.1002/mma.5281. [13] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. [14] M. Kwak, Finite-dimensional inertial forms for the 2D Navier-Stokes equations, Indiana Uni. Math. J., 41 (1992), 927-981.  doi: 10.1512/iumj.1992.41.41051. [15] A. J. Linot and M. D. Graham, Deep learning to discover and predict dynamics on an inertial manifold, Phys. Rev. E, 101 (2020), 8 pp. doi: 10.1103/physreve.101.062209. [16] N. V. Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equ. Operat. Theor., 32 (1998), 332-353.  doi: 10.1007/BF01203774. [17] J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications, Springer-Verlag Berlin (2003). [18] R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Progr. Nonlinear Differ. Equ. Appl., 50 (2002), 279-293. [19] A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, Berlin, 1983 doi: 10.1007/978-1-4612-5561-1. [20] R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee–Infante equation, J. Dynam. Differ. Equ., 15 (2003), 61-86.  doi: 10.1023/A:1026153311546. [21] G. R. Sell, Inertial manifolds: The non-self-adjoint case, J. Differ. Equ., 96 (1992), 203-255.  doi: 10.1016/0022-0396(92)90152-D. [22] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, 2002. doi: 10.1007/978-1-4757-5037-9. [23] S. Takagi, Smoothness of inertial manifolds for semilinear evolution equations in complex Banach spaces, Differ. Integral Equ., 21 (2008), 63-80. [24] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York 1988. doi: 10.1007/978-1-4684-0313-8. [25] S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proc. Roy. Soc. Edinb. Sec. A, 144 (2014), 1245-1327.  doi: 10.1017/S0308210513000073.

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##### References:
 [1] P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107. [2] P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differ. Integral Equ., 1 (1988), 433-457. [3] I. D. Chueshov and M. Scheutzow, Inertial manifolds and forms for stochastically perturbed retarded semilinear parabolic equations, J. Dynam. Differ. Equ., 13 (2001), 355-380.  doi: 10.1023/A:1016684108862. [4] P. Constantin, C. Foias, B. Nicolaenko and R. Temam, Integral Manifolds and Inertial Manifolds for Dissipative Partial Differential Equations, Springer, 1989. doi: 10.1007/BF01048790. [5] A. Debussche and R. Temam, Inertial manifolds and the slow manifolds in meteorology, Differ. Integral Equ., 4 (1991), 897-931. [6] A. Debussche and R. Temam, Some new generalizations of inertial manifolds, Discrete and Continuous Dynamical Systems, 2 (1996), 543-558.  doi: 10.3934/dcds.1996.2.543. [7] C. Foias, G. R. Sell, and R. Temam, Variétés inertielles des équations différentielles dissipatives, Comptes Rendus de l'Académie des Sciences - Series I - Mathematics, 301 (1985), 139–142. [8] K. Furuya and A. Yagi, Linearized stability for abstract quasilinear equations of parabolic type, Funkcial. Ekvac., 37 (1994), 483-504. [9] K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Grad. Texts in Math. 194, Springer, 2000. [10] N. T. Huy, Inertial Manifolds for Semi-linear Parabolic Equations in Admissible Spaces, J. Math. Anal. Appl., 386 (2012), 894-909.  doi: 10.1016/j.jmaa.2011.08.051. [11] N. T. Huy, Admissibly inertial manifolds for a class of semi-linear evolution equations, J. Differ. Equ., 254 (2013), 2638-2660.  doi: 10.4064/ap112-2-3. [12] N. T. Huy and X. Q. Bui, Competition models with diffusion, analytic semigroups, and inertial manifolds, Math. Method. Appl. Sci., 41 (2018), 8182-8200.  doi: 10.1002/mma.5281. [13] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. [14] M. Kwak, Finite-dimensional inertial forms for the 2D Navier-Stokes equations, Indiana Uni. Math. J., 41 (1992), 927-981.  doi: 10.1512/iumj.1992.41.41051. [15] A. J. Linot and M. D. Graham, Deep learning to discover and predict dynamics on an inertial manifold, Phys. Rev. E, 101 (2020), 8 pp. doi: 10.1103/physreve.101.062209. [16] N. V. Minh, F. Räbiger and R. Schnaubelt, Exponential stability, exponential expansiveness and exponential dichotomy of evolution equations on the half line, Integral Equ. Operat. Theor., 32 (1998), 332-353.  doi: 10.1007/BF01203774. [17] J. D. Murray, Mathematical Biology Ⅱ: Spatial Models and Biomedical Applications, Springer-Verlag Berlin (2003). [18] R. Nagel and G. Nickel, Well-posedness for non-autonomous abstract Cauchy problems, Progr. Nonlinear Differ. Equ. Appl., 50 (2002), 279-293. [19] A. Pazy, Semigroup of Linear Operators and Application to Partial Differential Equations, Springer-Verlag, Berlin, 1983 doi: 10.1007/978-1-4612-5561-1. [20] R. Rosa, Exact finite dimensional feedback control via inertial manifold theory with application to the Chafee–Infante equation, J. Dynam. Differ. Equ., 15 (2003), 61-86.  doi: 10.1023/A:1026153311546. [21] G. R. Sell, Inertial manifolds: The non-self-adjoint case, J. Differ. Equ., 96 (1992), 203-255.  doi: 10.1016/0022-0396(92)90152-D. [22] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, Springer-Verlag, 2002. doi: 10.1007/978-1-4757-5037-9. [23] S. Takagi, Smoothness of inertial manifolds for semilinear evolution equations in complex Banach spaces, Differ. Integral Equ., 21 (2008), 63-80. [24] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer, New York 1988. doi: 10.1007/978-1-4684-0313-8. [25] S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proc. Roy. Soc. Edinb. Sec. A, 144 (2014), 1245-1327.  doi: 10.1017/S0308210513000073.
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