Open Access
2002 Attractors and global averaging of non-autonomous reaction-diffusion equations in $\mathbb R^N$
Francesca Antoci, Martino Prizzi
Topol. Methods Nonlinear Anal. 20(2): 229-259 (2002).
Abstract

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 $\omega\to\infty$.

References

1.

F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains , Topol. Methods Nonlinear Anal., 18 (2001), 283–302 \ref  MR1911383F. Antoci and M. Prizzi, Reaction-diffusion equations on unbounded thin domains , Topol. Methods Nonlinear Anal., 18 (2001), 283–302 \ref  MR1911383

2.

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  MR1084733 10.1017/S0308210500031498A. 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  MR1084733 10.1017/S0308210500031498

3.

N. N. Bogolyubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York (1962) \ref N. N. Bogolyubov and Y. A. Mitropolski, Asymptotic Methods in the Theory of Nonlinear Oscillations, Gordon and Breach, New York (1962) \ref

4.

H. Brezis, Analyse Fonctionelle, Masson, Paris(1992) \ref H. Brezis, Analyse Fonctionelle, Masson, Paris(1992) \ref

5.

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 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

6.

A. Friedman, Partial Differential Equations, Robert E. Klieger Publishing Company, Malabar, Florida (1983) \ref  MR454266A. Friedman, Partial Differential Equations, Robert E. Klieger Publishing Company, Malabar, Florida (1983) \ref  MR454266

7.

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence(1988) \ref  MR941371J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys Monographs 25, Amer. Math. Soc., Providence(1988) \ref  MR941371

8.

J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions , J. Integral Equations Appl., 2 (1990), 463–494 \ref  MR1094480 10.1216/jiea/1181075583 euclid.jiea/1181075583 J. K. Hale and S. M. Verduyn Lunel, Averaging in infinite dimensions , J. Integral Equations Appl., 2 (1990), 463–494 \ref  MR1094480 10.1216/jiea/1181075583 euclid.jiea/1181075583

9.

A. Haraux, Attractors of asymptotically compact processes and applications to nonlinear partial differential equations , Comm. Partial Differential Equations, 13 (1988), 1383–1414 \ref  MR956826 10.1080/03605308808820580A. Haraux, Attractors of asymptotically compact processes and applications to nonlinear partial differential equations , Comm. Partial Differential Equations, 13 (1988), 1383–1414 \ref  MR956826 10.1080/03605308808820580

10.

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer–Verlag, New York (1981) \ref  MR610244D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, Vol. 840, Springer–Verlag, New York (1981) \ref  MR610244

11.

A. A. Ilyin, Global averaging of dissipative dynamical systems , Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), XXII (1998), 165–191 \ref  MR1695715A. A. Ilyin, Global averaging of dissipative dynamical systems , Rend. Accad. Naz. Sci. XL Mem. Mat. Appl. (5), XXII (1998), 165–191 \ref  MR1695715

12.

O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge (1991) \ref  MR1133627O. Ladyzhenskaya, Attractors for Semigroups and Evolution Equations, Cambridge University Press, Cambridge (1991) \ref  MR1133627

13.

B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge (1982) \ref  MR690064B. M. Levitan and V. V. Zhikov, Almost Periodic Functions and Differential Equations, Cambridge University Press, Cambridge (1982) \ref  MR690064

14.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York (1983) \ref  MR710486A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer–Verlag, New York (1983) \ref  MR710486

15.

M. Prizzi, A remark on reaction-diffusion equations in unbounded domains , DCDS-A, to appear \ref  MR1952374 10.3934/dcds.2003.9.281M. Prizzi, A remark on reaction-diffusion equations in unbounded domains , DCDS-A, to appear \ref  MR1952374 10.3934/dcds.2003.9.281

16.

G. R. Sell, Nonautonomous differential equations and topological dynamics \romI, \romII, Trans. Amer. Math. Soc., 127 (1967), 241–262, 263–284 \ref  MR212313G. R. Sell, Nonautonomous differential equations and topological dynamics \romI, \romII, Trans. Amer. Math. Soc., 127 (1967), 241–262, 263–284 \ref  MR212313

17.

H. Tanabe, Equations of Evolution, Pitman Press, Monographs and Studies in Mathematics 6, Bath (1979) \ref  MR533824H. Tanabe, Equations of Evolution, Pitman Press, Monographs and Studies in Mathematics 6, Bath (1979) \ref  MR533824

18.

R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York (1997) \ref  MR1441312R. Temam, Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer–Verlag, New York (1997) \ref  MR1441312

19.

B. Wang, Attractors for reaction-diffusion equations in unbounded domains , Physica D, 128 (1999), 41–52  MR1685247 10.1016/S0167-2789(98)00304-2B. Wang, Attractors for reaction-diffusion equations in unbounded domains , Physica D, 128 (1999), 41–52  MR1685247 10.1016/S0167-2789(98)00304-2
Copyright © 2002 Juliusz P. Schauder Centre for Nonlinear Studies
Francesca Antoci and Martino Prizzi "Attractors and global averaging of non-autonomous reaction-diffusion equations in $\mathbb R^N$," Topological Methods in Nonlinear Analysis 20(2), 229-259, (2002). https://doi.org/
Published: 2002
Vol.20 • No. 2 • 2002
Back to Top