DOI QR코드

DOI QR Code

ADMISSIBLE INERTIAL MANIFOLDS FOR INFINITE DELAY EVOLUTION EQUATIONS

  • Minh, Le Anh (Department of Mathematical Analysis Hong Duc University)
  • Received : 2020.05.23
  • Accepted : 2020.11.09
  • Published : 2021.05.31

Abstract

The aim of this paper is to prove the existence of an admissible inertial manifold for mild solutions to infinite delay evolution equation of the form $$\{{\frac{du}{dt}}+Au=F(t,\;u_t),\;t{\geq}s,\\\;u_s({\theta})={\phi}({\theta}),\;{\forall}{\theta}{\in}(-{{\infty}},\;0],\;s{\in}{\mathbb{R}},$$ where A is positive definite and self-adjoint with a discrete spectrum, the Lipschitz coefficient of the nonlinear part F may depend on time and belongs to some admissible function space defined on the whole line. The proof is based on the Lyapunov-Perron equation in combination with admissibility and duality estimates.

Keywords

Acknowledgement

The author would like to thank the anonymous referees who provided useful and detailed comments on the earlier versions of the manuscript.

References

  1. L. Boutet de Monvel, I. D. Chueshov, and A. V. Rezounenko, Inertial manifolds for retarded semilinear parabolic equations, Nonlinear Anal. 34 (1998), no. 6, 907-925. https://doi.org/10.1016/S0362-546X(97)00569-5
  2. I. D. Chueshov, Introduction to the Theory of Infinite-Dimensional Dissipative Systems, ACTA Scientific Publishing House, 2002.
  3. P. Constantin, C. Foias, B. Nicolaenko, and R. Temam, Integral manifolds and inertial manifolds for dissipative partial differential equations, Applied Mathematical Sciences, 70, Springer-Verlag, New York, 1989. https://doi.org/10.1007/978-1-4612-3506-4
  4. C. Foias, G. R. Sell, and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), no. 2, 309-353. https://doi.org/10.1016/0022-0396(88)90110-6
  5. N. T. Huy, Admissibly inertial manifolds for a class of semi-linear evolution equations, J. Differential Equations 254 (2013), 2638-2660. https://doi.org/10.1016/j.jde.2013.01.001
  6. N. Koksch and S. Siegmund, Pullback attracting inertial manifolds for nonautonomous dynamical systems, J. Dynam. Differential Equations 14 (2002), no. 4, 889-941. https://doi.org/10.1023/A:1020768711975
  7. T. H. Nguyen and A. M. Le, Admissible inertial manifolds for delay equations and applications to Fisher-Kolmogorov model, Acta Appl. Math. 156 (2018), 15-31. https://doi.org/10.1007/s10440-017-0153-y