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Dynamics of periodically forced parabolic equations on the circle

Published online by Cambridge University Press:  19 September 2008

Björn Sandstede  and
Bernold Fiedler
Björn Sandstede
Affiliation:
Mathematics Institute A, University of Stuttgart, Pfoffenwaldring 57, 7000 Stuttgart 80, Germany
Bernold Fiedler
Affiliation:
Mathematics Institute A, University of Stuttgart, Pfoffenwaldring 57, 7000 Stuttgart 80, Germany
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Abstract

We consider the dynamics of scalar reaction-diffusion equations:

We show that every prescribed vectorfield in the plane embeds into equation (1) for a suitable f = f (t, x, u, ux). In contrast, suppose f = f (t, u, ux) is independent of x and T-periodic in t. Then the ω-limit set of any bounded solution is a subset of some two-dimensional torus carrying a linear flow.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 12 , Issue 3 , September 1992 , pp. 559 - 571
Copyright
Copyright © Cambridge University Press 1992

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References

REFERENCES

Angenent, S. (1988), The zero set of a solution of a parabolic equation. J. Reine Angew. Math. 390 79–96.Google Scholar
Angenent, S. & Fiedler, B. (1988), The dynamics of rotating waves in scalar reaction diffusion equations. Trans. Amer. Math. Soc. 307 545–568.CrossRefGoogle Scholar
Brunovský, P., Poláčik, P. & Sandstede, B. (1992), Convergence in general parabolic equations in one space dimension. Nonlin. Analysis TMA 18 209–215.CrossRefGoogle Scholar
Coddington, E. A. & Levinson, N. (1955), Theory of Ordinary Differential Equations. McGraw-Hill, New York.Google Scholar
Chen, X.-Y. & Matano, H. (1989), Convergence, asymptotic periodicity and finite point blow-up in one-dimensional heat equations. J. Diff. Eq. 78 160–190.CrossRefGoogle Scholar
Cornfeld, I. P., Fomin, S. V. & Sinai, Ya. G. (1982), Ergodic Theory. Springer-Verlag, New York.CrossRefGoogle Scholar
Dancer, E. N. (1990), On the existence of two-dimensional invariant tori for scalar parabolic equations with time periodic coefficients. Ann. Sc. Norm. Sup. Pisa 18 (1991), 455–471.Google Scholar
Fiedler, B. & Mallet-Paret, J. (1989), A Poincaré-Bendixson theorem for scalar reaction diffusion equations. Arch. Rational Mech. Analysis 107 325–345.CrossRefGoogle Scholar
Henry, D. (1981), Geometric theory of semilinear parabolic equations. Springer Lecture Notes in Mathematics 840. Springer-Verlag, New York.CrossRefGoogle Scholar
Massatt, P. (1986), The convergence of solutions of scalar reaction diffusion equations with convection to periodic solutions. Preprint.Google Scholar
Matano, H. (1986), Asymptotic behavior of solutions of semilinear heat equations on the circle. In Nonlinear Diffusion Equations and their Equilibrium States. Ni, W. -M., Peletier, L. A., Serrin, J., eds, Proc. Berkeley, Springer-Verlag, New York.Google Scholar
Nadirashvili, N. (1990), On the dynamics of nonlinear parabolic equations. Sov. Math. Dokl. 40 no 3, 636–639.Google Scholar
Poláčik, P. (1992), Imbedding of any vector field in a scalar semilinear parabolic equation. To appear in Proc Amer. Math. Soc.CrossRefGoogle Scholar
Sandstede, B. (1992), Asymptotic behavior of solutions of non-autonomous scalar reaction-diffusion equations. Submitted to Proceedings of Equadiff-91. Perello, C., Simo, C. and Sola-Morales, J., eds, World Scientific.Google Scholar