MR2108675
Masashi AIDA and Atsushi YAGI
Masashi AIDA and Atsushi YAGI
Global Stability of Approximation for Exponential Attractors
Funkcialaj Ekvacioj. Serio Internacia
47
2004
251--276
http://fe.math.kobe-u.ac.jp/FE/FullPapers/47-2/47_251.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2108675
This paper is concerned with the initial value problem for some diffusion system which describes the process of a pattern formation of biological individuals by chemotaixis and growth. In the paper Osaki et al. [13], exponential attractors have been constructed for the dynamical system determined by this problem. The exponential attractor is one of limit sets which is a positively invariant compact set with finite fractal dimension and which attracts every trajectory in an exponential rate. In this paper we study another feature of exponential attractors, that is we show that the approximate solution also gets close to the exponential attractor in an exponential rate and remains in its neighborhood forever. Our methods are available to any other exponential attractors determined by interaction-diffusion systems.
Exponential attractors, Global stability under approximation, Chemotaxis-growth system.
Primary 37L15; Secondary 65P99, 92D25.
47-251
2004
Global Stability of Approximation for Exponential Attractors
Masashi AIDA and Atsushi YAGI
Masashi AIDA and Atsushi YAGI
1
Aida, M.; Yagi, A.
Global attractor for approximate system of chemotaxis and growth
Dynam. Conti. Discrete Impluls. Systems Series A
10
2003
309-315
MR1974252
2
Aida, M.; Efendiev, M.; Yagi, A., Exponential attractor for quasilinear parabolic evolution equations, Osaka J. Math., to appear
MR2132006
3
Alt, W.; Lauffenburger, D. A.
Transient behavior of a chemotaxis system modelling certain types of tissue inflammation
J. Math. Biol.
24
1985
691-722
MR0880453
4
Budrene, E. O.; Berg, H. C.
Complex patterns formed by motile cells of Escherichia coli
Nature
349
1991
630-633
5
Efendiev, M.; Miranville, A.; Zelik, S.
Exponential attractors for a nonlinear reaction-diffusion systems in $R^3$
C. R. Acad. Sci. Paris Série I
330
2000
713-718
MR1763916
6
Ford, R. M.; Lauffenburger, D. A.
Analysis of chemotactic bacterial distributions in population migration assays using a mathematical model applicable to steep or shallow attractant gradients
Bull. Math. Biol.
53
1991
721-749
7
Lauffenburger, D. A.; Kennedy, C. R.
Localized bacterial infection in a distributed model for tissue inflammation
J. Math. Biol.
16
1983
141-163
8
Mimura, M.; Tsujikawa, T.
Aggregating pattern dynamics in a chemotaxis model including growth
Physica A
230
1996
499-543
9
Mimura, M.; Tsujikawa, T.; Kobayashi, R.; Ueyama, D.
Dynamics of aggregating patterns in a chemotaxis-diffusion-growth model equation
Forum
8
1993
179-195
MR1483384
10
Myerscough, M. R.; Murray, J. D.
Analysis of propagating pattern in a chemotaxis system
Bull. Math. Biol.
54
1992
77-94
11
Nakaguchi, E.; Yagi, A.
Fully discrete approximation by Galerkin Runge-Kutta methods for quasilinear parabolic systems
Hokkaido J. Math.
33
2002
385-429
MR1914967
12
Nakaguchi, E.; Yagi, A.
Full discrete approximations by Galerkin method for chemotaxis-growth model
Proc. WCNA2000, Nonlinear Analysis
47
2001
6097-6107
MR1970781
13
Osaki, K.; Tsujikawa, T.; Yagi, A.; Mimura, M.
Exponential attractor for a chemotaxis-growth system of equations
Nonlinear Analysis
51
2002
119-144
MR1915744
14
Ryu, S.-U.; Yagi, A.
Optimal control of Keller-Segel equations
J. Math. Anal. Appl.
256
2001
45-66
MR1820067
15
Tsujikawa, T.
Singular limit analysis of planar equilibrium solutions to a chemotaxis model equation with growth
Methods Applications Anal.
3
1996
401-431
MR1437787
16
Tyson, R.; Stern, L. G.; LeVeque, R. J.
Fractional step methods supplied to a chemotaxis model
J. Math. Biol.
41
2000
455-475
MR1803855
17
Woodward, D. E.; Tyson, R.; Myerscough, M. R.; Murray, J. D.; Budrene, E. O.; Berg, H. C.
Spatio-temporal patterns generated by Salmonella typhimurium
Biophys. J.
68
1995
2181-2189
18
Yagi, A.
Parabolic evolution equations in which the coefficients are the generators of infinitely differentially semigroups
Funkcial. Ekvac.
32
1989
107-124
MR1006090
1006090
19
Yagi, A.
Parabolic evolution equations in which the coefficients are the generators of infinitely differentially semigroups, II
Funkcial. Ekvac.
33
1990
139-150
MR1065472
1065472
20
Yagi, A.
Abstract quasilinear evolution equations of parabolic type in Banach spaces
Boll. Un. Mat. Ital.
5-B
1991
341-368
MR1111127
21
Yagi, A.
Quasilinear abstract parabolic evolution equations with applications
"Evolution Equations, Semigroups and Functional Analysis", eds. A. Lorenzi and B. Ruf, Birkhäuser, Verlag Basel
2002
381-397
MR1944173
22
Eden, A.; Foias, C.; Nicolaenko, B.; Temam, R.
Exponential Attractors for Dissipative Evolution Equations
John Wiley & Sons, Chichester, New York
1994
MR1335230
23
Haken, H.
Synergetics, An Introduction 3rd ed.
Springer-Verlag, New York, Berlin, Heidelberg
1983
MR0714329
24
Nicolis, G.; Prigogine, I.
Self-Organization in Nonequilibrium System-From Dissipative Structure to Order through Fluctuations
John Wiley & Sons, Chichester, New York
1997
MR0522141
25
Temam, R.
Infinite-Dimensional Dynamical systems in Mechanics and Physics 2nd ed.
Springer-Verlag, New York, Belin, Heidelberg
1997
MR1441312