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Periodic solutions and Galerkin approximations to the autonomous reaction-diffusion equations

Published online by Cambridge University Press:  17 April 2009

Arnold Dikansky
Arnold Dikansky
Affiliation:
Div of Mathematics and Computer Science, St John’s University, 300 Howard Ave Staten Island NY 10301, United States of America
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The assumption that a Galerkin equation of the reaction-diffusion system of high order has an asymptotically orbitally stable time-periodic solution implies that the full reaction-diffusion system has a nearby asymptotically orbitally stable time-periodic solution with asymptotic phase.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 48 , Issue 1 , August 1993 , pp. 127 - 140
Copyright
Copyright © Australian Mathematical Society 1993

References

[1]Alikakos, N.D., ‘An application of the invariance principle to reaction-diffusion equations’, J. Differential Equations 33 (1979), 201–225.CrossRefGoogle Scholar
[2]Cesari, L., Functional analysis and periodic solutions to nonlinear differential equations, Vol. 1 (Interscience, New York, 1962).Google Scholar
[3]Constantin, P., Foias, C. and Temam, R., ‘On the large time Galerkin approximation of the Navier-Stokes equations’, SIAM J. Numer. Anal. 21 (1984), 615–634.CrossRefGoogle Scholar
[4]Dikansky, A., ‘On the large time behavior of Galerkin approximations to the reaction-diffusion equations’, Numer. Funct. Anal. Optim. (Submitted)Google Scholar
[5]Dikansky, A., ‘Asymptotically stable stationary solutions of reaction-diffusion equations and their Galerkin approximations’, Bull. Austral. Math. Soc. 47 (1993), 271–284.Google Scholar
[6]Hale, J., Ordinary differential equations (Wiley-Interscience, New York, 1969).Google Scholar
[7]Henry, D., Geometrical theory of semilinear parabolic equations, Lecture Notes in Math. 840 (Springer-Verlag, Berlin, Heidelberg, New York, 1981).CrossRefGoogle Scholar
[8]Kloden, P.E., ‘Asymptotically stable solutions of the Navier-Stokes equations and its Galerkin approximations’, in Miniconference on Nonlinear Analysis (Canberra, 1984), Proc. Centre Math. Anal. Austral. Nat. Univ. 8 (Australian National University, Canberra), pp. 137–150.Google Scholar
[9]Kloedon, P.E., ‘Asymptotically stable attracting sets for the Navier-Stokes equations’, Bull. Austral. Math. Soc. 34 (1986), 37–52.CrossRefGoogle Scholar
[10]Stokes, A., ‘On the approximation of nonlinear oscillations’, J. Differential Equations 12 (1972), 535–558.Google Scholar
[11]Titi, E.S., ‘On a criterion for location stable stationary solutions to the Navier-Stokes equations’, Nonlinear Anal. 11 (1987), 1085–1102.Google Scholar
[12]Titi, E.S., ‘Critére numérique pour mettre en évidence des solutions periodiques des équations de Navier-Stokes’, C.R. Acad. Sci. Paris Ser. I (1990).Google Scholar