A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations
HTML articles powered by AMS MathViewer
- by Fengbo Hang and Huiqiang Jiang PDF
- Proc. Amer. Math. Soc. 134 (2006), 2633-2637 Request permission
Abstract:
In 1992, P. Poláčik showed that one could linearly imbed any vector field into a scalar semi-linear parabolic equation on $\Omega$ with Neumann boundary condition provided that there exists a smooth vector field $\Phi =\left ( \phi _{1},\cdots ,\phi _{n}\right )$ on $\overline {\Omega }$ such that \[ \left \{ \begin {array} [c]{l} \operatorname {rank}\left ( \Phi \left ( x\right ) ,\partial _{1}\Phi \left ( x\right ) ,\cdots ,\partial _{n}\Phi \left ( x\right ) \right ) =n\text { for all }x\in \overline {\Omega }, \frac {\partial \Phi }{\partial \nu }=0\text { on }\partial \Omega \text {.} \end {array} \right . \] In this short paper, we give a classification of all the domains on which one may find such a type of vector field.References
- Herbert Amann, Existence and regularity for semilinear parabolic evolution equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 11 (1984), no. 4, 593–676. MR 808425
- Glen E. Bredon, Topology and geometry, Graduate Texts in Mathematics, vol. 139, Springer-Verlag, New York, 1997. Corrected third printing of the 1993 original. MR 1700700, DOI 10.1007/978-1-4612-0647-7
- E. N. Dancer and P. Poláčik, Realization of vector fields and dynamics of spatially homogeneous parabolic equations, Mem. Amer. Math. Soc. 140 (1999), no. 668, viii+82. MR 1618487, DOI 10.1090/memo/0668
- Daniel Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR 610244, DOI 10.1007/BFb0089647
- John W. Milnor, Topology from the differentiable viewpoint, Princeton Landmarks in Mathematics, Princeton University Press, Princeton, NJ, 1997. Based on notes by David W. Weaver; Revised reprint of the 1965 original. MR 1487640
- Peter Poláčik, Complicated dynamics in scalar semilinear parabolic equations in higher space dimension, J. Differential Equations 89 (1991), no. 2, 244–271. MR 1091478, DOI 10.1016/0022-0396(91)90121-O
- P. Poláčik, Imbedding of any vector field in a scalar semilinear parabolic equation, Proc. Amer. Math. Soc. 115 (1992), no. 4, 1001–1008. MR 1089411, DOI 10.1090/S0002-9939-1992-1089411-7
- Peter Poláčik, High-dimensional $\omega$-limit sets and chaos in scalar parabolic equations, J. Differential Equations 119 (1995), no. 1, 24–53. MR 1334487, DOI 10.1006/jdeq.1995.1083
- Peter Poláčik, Reaction-diffusion equations and realization of gradient vector fields, International Conference on Differential Equations (Lisboa, 1995) World Sci. Publ., River Edge, NJ, 1998, pp. 197–206. MR 1639355
- P. Poláčik, Parabolic equations: asymptotic behavior and dynamics on invariant manifolds, Handbook of dynamical systems, Vol. 2, North-Holland, Amsterdam, 2002, pp. 835–883. MR 1901067, DOI 10.1016/S1874-575X(02)80037-6
- Peter Poláčik and Krzysztof Rybakowski, Imbedding vector fields in scalar parabolic Dirichlet BVPs, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22 (1995), no. 4, 737–749. MR 1375317
- M. Prizzi, Perturbation of elliptic operators and complex dynamics of parabolic partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 130 (2000), no. 2, 397–418. MR 1750838, DOI 10.1017/S0308210500000226
- Martino Prizzi and Krzysztof P. Rybakowski, Some recent results on chaotic dynamics of parabolic equations, Proceedings of the Conference “Topological Methods in Differential Equations and Dynamical Systems” (Kraków-Przegorzały, 1996), 1998, pp. 231–235. MR 1661351
- Martino Prizzi and Krzysztof P. Rybakowski, Inverse problems and chaotic dynamics of parabolic equations on arbitrary spatial domains, J. Differential Equations 142 (1998), no. 1, 17–53. MR 1492876, DOI 10.1006/jdeq.1997.3338
- Martino Prizzi and Krzysztof P. Rybakowski, Complicated dynamics of parabolic equations with simple gradient dependence, Trans. Amer. Math. Soc. 350 (1998), no. 8, 3119–3130. MR 1491875, DOI 10.1090/S0002-9947-98-02294-6
- Krzysztof P. Rybakowski, The center manifold technique and complex dynamics of parabolic equations, Topological methods in differential equations and inclusions (Montreal, PQ, 1994) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 472, Kluwer Acad. Publ., Dordrecht, 1995, pp. 411–446. MR 1368677
Additional Information
- Fengbo Hang
- Affiliation: Department of Mathematics, Michigan State University, East Lansing, Michigan 48824
- Email: fhang@math.msu.edu
- Huiqiang Jiang
- Affiliation: School of Mathematics, University of Minnesota, 127 Vincent Hall, 206 Church St. S.E., Minneapolis, Minnesota 55455
- Email: hqjiang@math.umn.edu
- Received by editor(s): March 25, 2005
- Published electronically: April 7, 2006
- Additional Notes: The research of the first author was supported in part by NSF Grant DMS-0209504.
- Communicated by: David S. Tartakoff
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 2633-2637
- MSC (2000): Primary 35K20; Secondary 35B40, 54F65
- DOI: https://doi.org/10.1090/S0002-9939-06-08627-8
- MathSciNet review: 2213742