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A remark on the existence of suitable vector fields related to the dynamics of scalar semi-linear parabolic equations


Authors: Fengbo Hang and Huiqiang Jiang
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
Published electronically: April 7, 2006
MathSciNet review: 2213742
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Abstract: In 1992, P. Polácik 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

\begin{displaymath} \left\{ \begin{array}[c]{l} \operatorname*{rank}\left( \Phi\... ...tial\nu}=0\text{ on }\partial\Omega\text{.} \end{array}\right. \end{displaymath}

In this short paper, we give a classification of all the domains on which one may find such a type of vector field.


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  • [A] H. Amann. Existence and regularity for semilinear parabolic evolution equations. Scuola Morm. Sup. Pisa Cl. Sci. 11 (4):593-696, 1984. MR 0808425 (87h:34088)
  • [B] G. E. Bredon. Topology and geometry. Graduate Texts in Mathematics, 139. Springer-Verlag, New York, 1997. MR 1700700 (2000b:55001)
  • [DP] E. N. Dancer and P. Polácik. Realization of vector fields and dynamics of spatially homogeneous parabolic equations. Mem. AMS 140(668), 1999. MR 1618487 (99m:35125)
  • [H] D. Henry. Geometric theory of semilinear parabolic equations. Lecture notes in mathematics, Vol. 840, Springer-Verlag, New York, 1983. MR 0610244 (83j:35084)
  • [M] J. W. Milnor. Topology from the differentiable viewpoint. Princeton Landmarks in Mathematics. Princeton University Press, Princeton, NJ, 1997. MR 1487640 (98h:57051)
  • [P1] P. Polácik. Complicated dynamics in scalar semilinear parabolic equations in higher space dimension. Journal of Differential Equations 89(2):244-271, 1991. MR 1091478 (92c:35063)
  • [P2] P. Polácik. Imbedding of any vector field in a scalar semilinear parabolic equation. Proc. AMS. 115(4):1001-1008, 1992. MR 1089411 (92j:35099)
  • [P3] P. Polácik. High-dimensional $ \omega$-limit sets and chaos in scalar parabolic equations. J. Differential Equations 119(1):24-53, 1995. MR 1334487 (96h:35092)
  • [P4] P. Polácik. Reaction-diffusion equations and realization of gradient vector fields. In International Conference on Differential Equations (Lisboa, 1995), 197-206. World Scientific Publishing, River Edge, NJ, 1998. MR 1639355 (99j:35109)
  • [P5] P. Polácik. Parabolic equations: asymptotic behavior and dynamics on invariant manifolds. Handbook on Dynamical Systems vol. 2: 835-883, B. Fiedler (ed.), Elsevier, Amsterdam, 2002. MR 1901067 (2003f:37150)
  • [PR] P. Polácik and K. P. Rybakowski. Imbedding vector fields in scalar parabolic Dirichlet BVPs. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 22(4):737-749, 1995. MR 1375317 (97a:35124)
  • [Pr] M. Prizzi. Perturbation of elliptic operators and complex dynamics of parabolic partial differential equations. Proc. Roy. Soc. Edinburgh Sect. A 130(2):397-418, 2000. MR 1750838 (2001e:35096)
  • [PrR1] M. Prizzi and K. P. Rybakowski. Some recent results on chaotic dynamics of parabolic equations. In Proceedings of the Conference ``Topological Methods in Differential Equations and Dynamical Systems'' (Kraków-Przegorzaly, 1996), no. 36, 231-235, 1998. MR 1661351
  • [PrR2] M. Prizzi and K. P. Rybakowski. Inverse problems and chaotic dynamics of parabolic equations on arbitrary spatial domains. J. Differential Equations 142(1):17-53, 1998. MR 1492876 (98k:35207)
  • [PrR3] M. Prizzi and K. P. Rybakowski. Complicated dynamics of parabolic equations with simple gradient dependence. Trans. Amer. Math. Soc. 350(8):3119-3130, 1998. MR 1491875 (99a:35125)
  • [R] K. P. Rybakowski. The center manifold technique and complex dynamics of parabolic equations. In Topological methods in differential equations and inclusions (Montreal, PQ, 1994), Volume 472 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. 411-446. Kluwer Acad. Publ., Dordrecht, 1995. MR 1368677 (97b:35095)

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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

DOI: https://doi.org/10.1090/S0002-9939-06-08627-8
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
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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