Skip to main content Skip to main navigation menu Skip to site footer
  • Login
  • Language
    • English
    • Język Polski
  • Menu
  • Home
  • Current
  • Online First
  • Archives
  • About
    • About the Journal
    • Submissions
    • Editorial Team
    • Privacy Statement
    • Contact
  • Login
  • Language:
  • English
  • Język Polski

Topological Methods in Nonlinear Analysis

$L^{p}$-pullback attractors for non-autonomous reaction-diffusion equations with delays
  • Home
  • /
  • $L^{p}$-pullback attractors for non-autonomous reaction-diffusion equations with delays
  1. Home /
  2. Archives /
  3. Vol 54, No 1 (September 2019) /
  4. Articles

$L^{p}$-pullback attractors for non-autonomous reaction-diffusion equations with delays

Authors

  • Kaixuan Zhu
  • Yongqin Xie
  • Feng Zhou

Keywords

Reaction-diffusion equations, delays, pullback attractors

Abstract

In this paper, we consider the non-autonomous reaction-diffusion equations with hereditary effects and the nonlinear term $f$ satisfying the polynomial growth of arbitrary order $p-1$ $(p\geq2)$. The delay term may be driven by a function with very weak assumptions, namely, just measurability. We extend the asymptotic \emph{a priori} estimate method (see \cite{ZYS06}) to our problem and establish a new existence theorem for the pullback attractors in $C_{L^{p}(\Omega)}$ $(p> 2)$ (see Theorem \ref{t2.5}), which generalizes the results obtained in \cite{GM14}.

References

A.V. Babin and M.I. Vishik, Attractors of Evolution Equations, North-Holland, Amsterdam, 1992.

J.M. Ball, Global attractors for damped semilinear wave equations, Discrete Contin. Dyn. Syst. 10 (2004), 31–52.

T. Caraballo, X.Y. Han and P.E. Kloeden, Nonautonomous chemostats with variable delays, SIAM J. Math. Anal. 47 (2015), 2178–2199.

T. Caraballo, P.E. Kloeden and P. Marı́n-Rubio, Numerical and finite delay approximations of attractors for logistic differential-integral equations with infinite delay, Discrete Contin. Dyn. Syst. 19 (2007), 177–196.

A.N. Carvalho, J.A. Langa and J.C. Robinson, Attractors for Infinite-Dimensional Non-autonomous Dynamical Systems, Appl. Math. Sci., Vol. 182, Springer, New York, 2013.

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for asymptotically compact non-autonomous dynamical systems, Nonlinear Anal. 64 (2006) 484–498.

T. Caraballo, G. Lukaszewicz and J. Real, Pullback attractors for non-autonomous 2D-Navier–Stokes equations in some unbounded domains, C. R. Math. Acad. Sci. Paris 342 (2006), 263–268.

T. Caraballo, P. Marı́n-Rubio and J. Valero, Autonomous and non-autonomous attractors for differential equations with delays, J. Differential Equations 208 (2005), 9–41.

T. Caraballo, P. Marı́n-Rubio and J. Valero, Attractors for differential equations with unbounded delays, J. Differential Equations 239 (2007), 311–342.

T. Caraballo and J. Real, Attractors for 2D-Navier–Stokes models with delays, J. Differential Equations 205 (2004), 271–297.

T. Caraballo, J. Real and A.M. Márquez, Three-dimensional system of globally modified Navier–Stokes equations with delay, Internat. J. Bifur. Chaos Appl. Sci. Engrg. 20 (2010), 2869–2883.

J. Garcı́a-Luengo and P. Marı́n-Rubio, Reaction-diffusion equations with nonautonomous force in H −1 and delays under measurability conditions on the driving delay term, J. Math. Anal. Appl. 417 (2014), 80–95.

J. Garcı́a-Luengo, P. Marı́n-Rubio and G. Planas, Attractors for a double timedelayed 2D-Navier–Stokes model, Discrete Contin. Dyn. Syst. 34 (2014), 4085–4105.

J. Garcı́a-Luengo, P. Marı́n-Rubio and J. Real, Pullback attractors for 2D Navier–Stokes equations with delays and their regularity, Adv. Nonlinear Stud. 13 (2013), 331–357.

J. Garcı́a-Luengo, P. Marı́n-Rubio and J. Real, Regularity of pullback attractors and attraction in H 1 in arbitrarily large finite intervals for 2D Navier–Stokes equations with infinite delay, Discrete Contin. Dyn. Syst. 34 (2014), 181–201.

J. Garcı́a-Luengo, P. Marı́n-Rubio and J. Real, Some new regularity results of pullback attractors for 2D Navier–Stokes equations with delays, Commun. Pure Appl. Anal. 14 (2015), 1603–1621.

J.K. Hale, Asymptotic Behavior of Dissipative Systems, Amer. Math. Soc., Providence, RI, 1988.

J.K. Hale and S.M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer–Verlag, New York, 1993.

P.E. Kloeden, Upper semi continuity of attractors of delay differential equations in the delay, Bull. Austral. Math. Soc. 73 (2006), 299–306.

P.E. Kloeden and P. Marı́n-Rubio, Equi-attraction and the continuous dependence of attractors on time delays, Discrete Contin. Dyn. Syst. Ser. B 9 (2008), 581–593.

G. Lukaszewicz, On pullback attractors in Lp for nonautonomous reaction-diffusion equations, Nonlinear Anal. 73 (2010), 350–357.

G. Lukaszewicz, On pullback attractors in H01 for nonautonomous reaction-diffusion equations, Internat. J. Bifur. Chaos 20 (2010), 2637–2644.

P. Marı́n-Rubio, A.M. Márquez-Durán and J. Real, Three dimensional system of globally modified Navier–Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. Ser. B 14 (2010), 655–673.

P. Marı́n-Rubio, A.M. Márquez-Durán and J. Real, Pullback attractors for globally modified Navier–Stokes equations with infinite delays, Discrete Contin. Dyn. Syst. 31 (2011), 779–796.

J.C. Robinson, Infinite-Dimensional Dynamical Systems: An Introduction to Dissipative Parabolic PDEs and the Theory of Global Attractors, Cambridge, Cambridge University Press, 2001.

R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York, 1997.

Y.J. Wang and P.E. Kloeden, The uniform attractor of a multi-valued process generated by reaction-diffusion delay equations on an unbounded domain, Discrete Contin. Dyn. Syst. 34 (2014), 4343–4370.

F. Wu and P.E. Kloeden, Mean-square random attractors of stochastic delay differential equations with random delay, Discrete Contin. Dyn. Syst. Ser. B 18 (2013), 1715–1734.

C.K. Zhong, M.H. Yang and C.Y. Sun, The existence of global attractors for the normto-weak continuous semigroup and application to the nonlinear reaction-diffusion equations, J. Differential Equations 223 (2006), 367–399.

Downloads

  • PREVIEW
  • FULL TEXT

Published

2019-06-20

How to Cite

1.
ZHU, Kaixuan, XIE, Yongqin and ZHOU, Feng. $L^{p}$-pullback attractors for non-autonomous reaction-diffusion equations with delays. Topological Methods in Nonlinear Analysis. Online. 20 June 2019. Vol. 54, no. 1, pp. 9 - 27. [Accessed 5 July 2025].
  • ISO 690
  • ACM
  • ACS
  • APA
  • ABNT
  • Chicago
  • Harvard
  • IEEE
  • MLA
  • Turabian
  • Vancouver
Download Citation
  • Endnote/Zotero/Mendeley (RIS)
  • BibTeX

Issue

Vol 54, No 1 (September 2019)

Section

Articles

Stats

Number of views and downloads: 0
Number of citations: 0

Search

Search

Browse

  • Browse Author Index
  • Issue archive

User

User

Current Issue

  • Atom logo
  • RSS2 logo
  • RSS1 logo

Newsletter

Subscribe Unsubscribe
Up

Akademicka Platforma Czasopism

Najlepsze czasopisma naukowe i akademickie w jednym miejscu

apcz.umk.pl

Partners

  • Akademia Ignatianum w Krakowie
  • Akademickie Towarzystwo Andragogiczne
  • Fundacja Copernicus na rzecz Rozwoju Badań Naukowych
  • Instytut Historii im. Tadeusza Manteuffla Polskiej Akademii Nauk
  • Instytut Kultur Śródziemnomorskich i Orientalnych PAN
  • Instytut Tomistyczny
  • Karmelitański Instytut Duchowości w Krakowie
  • Ministerstwo Kultury i Dziedzictwa Narodowego
  • Państwowa Akademia Nauk Stosowanych w Krośnie
  • Państwowa Akademia Nauk Stosowanych we Włocławku
  • Państwowa Wyższa Szkoła Zawodowa im. Stanisława Pigonia w Krośnie
  • Polska Fundacja Przemysłu Kosmicznego
  • Polskie Towarzystwo Ekonomiczne
  • Polskie Towarzystwo Ludoznawcze
  • Towarzystwo Miłośników Torunia
  • Towarzystwo Naukowe w Toruniu
  • Uniwersytet im. Adama Mickiewicza w Poznaniu
  • Uniwersytet Komisji Edukacji Narodowej w Krakowie
  • Uniwersytet Mikołaja Kopernika
  • Uniwersytet w Białymstoku
  • Uniwersytet Warszawski
  • Wojewódzka Biblioteka Publiczna - Książnica Kopernikańska
  • Wyższe Seminarium Duchowne w Pelplinie / Wydawnictwo Diecezjalne „Bernardinum" w Pelplinie

© 2021- Nicolaus Copernicus University Accessibility statement Shop