Topological Methods in Nonlinear Analysis

Attractors and global averaging of non-autonomous reaction-diffusion equations in $\mathbb R^N$

Francesca Antoci and Martino Prizzi

Full-text: Open access


We consider a family of non-autonomous reaction-diffusion equations $$ u_t=\sum_{i,j=1}^N a_{ij}(\omega t)\partial_i\partial_j u+f(\omega t,u)+ g(\omega t,x), \quad x\in\mathbb R^N \tag{$\text{\rm E}_\omega$} $$ with almost periodic, rapidly oscillating principal part and nonlinear interactions. As $\omega\to \infty$, we prove that the solutions of $(\text{\rm E}_\omega)$ converge to the solutions of the averaged equation $$ u_t=\sum_{i,j=1}^N \overline a_{ij}\partial_i\partial_j u+\overline f(u)+ \overline g(x), \quad x\in\mathbb R^N. \tag{$\text{\rm E}_\infty$} $$ If $f$ is dissipative, we prove existence and upper-semicontinuity of attractors for the family (E$_\omega$) as $\omega\to\infty$.

Article information

Topol. Methods Nonlinear Anal., Volume 20, Number 2 (2002), 229-259.

First available in Project Euclid: 1 August 2016

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Antoci, Francesca; Prizzi, Martino. Attractors and global averaging of non-autonomous reaction-diffusion equations in $\mathbb R^N$. Topol. Methods Nonlinear Anal. 20 (2002), no. 2, 229--259.

Export citation


  • F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains , Topol. Methods Nonlinear Anal., 18 (2001), 283–302 \ref
  • A. V. Babin and M. I. Vishik, Attractors of Evolution Equations, North Holland, Amsterdam (1991) \ref ––––, Attractors of partial differential evolution equations in an unbounded domain , Proc. Roy. Soc. Edinburgh Sect. A, 116 (1990), 221–243 \ref
  • N. N. Bogolyubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York (1962) \ref
  • H. Brezis, Analyse Fonctionelle, Masson, Paris(1992) \ref
  • V. V. Chepyzhov and M. I. Vishik, Attractors of non-autonomous dynamical systems and their dimension , J. Math. Pures Appl., 73 (1994), 279–333 \ref
  • A. Friedman, Partial Differential Equations, Robert E. Klieger Publishing Company, Malabar, Florida (1983) \ref
  • J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence(1988) \ref
  • J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions , J. Integral Equations Appl., 2 (1990), 463–494 \ref
  • A. Haraux, Attractors of asymptotically compact processes and applications to nonlinear partial differential equations , Comm. Partial Differential Equations, 13 (1988), 1383–1414 \ref
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer–Verlag, New York (1981) \ref
  • A. A. Ilyin, Global averaging of dissipative dynamical systems , Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), XXII (1998), 165–191 \ref
  • O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge (1991) \ref
  • B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge (1982) \ref
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York (1983) \ref
  • M. Prizzi, A remark on reaction-diffusion equations in unbounded domains , DCDS-A, to appear \ref
  • G. R. Sell, Nonautonomous differential equations and topological dynamics \romI, \romII, Trans. Amer. Math. Soc., 127 (1967), 241–262, 263–284 \ref
  • H. Tanabe, Equations of Evolution, Pitman Press, Monographs and Studies in Mathematics 6, Bath (1979) \ref
  • R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York (1997) \ref
  • B. Wang, Attractors for reaction-diffusion equations in unbounded domains , Physica D, 128 (1999), 41–52