Imbedding of any vector field in a scalar semilinear parabolic equation
Author:
P. Poláčik
Journal:
Proc. Amer. Math. Soc. 115 (1992), 10011008
MSC:
Primary 35K60
MathSciNet review:
1089411
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Abstract: The scalar semilinear parabolic equation on a smooth bounded convex domain under Neumann boundary condition (2) is considered. For any prescribed vector field on , a function is found such that the flow of (1), (2) has an invariant dimensional subspace and the vector field generating the flow of (1), (2) on this invariant subspace coincides, in appropriate coordinates, with .
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 D. Gilbarg and Trudinger, Elliptic partial differential equations of second order, SpringerVerlag, Berlin, 1977. MR 0473443 (57:13109)
 [GH]
 J. Guckenheimer and P. Holmes, Nonlinear oscillations, dynamical systems, and bifurcations of vector fields, SpringerVerlag, New York, 1983. MR 709768 (85f:58002)
 [He1]
 D. Henry, Geometric theory of semilinear parabolic equations, Lecture Notes in Math., vol. 840, SpringerVerlag, New York, 1981. MR 610244 (83j:35084)
 [He2]
 , Perturbation of the boundary value problems for partial differential equations, Seminaire Brasileiro de Analise, Trabalhos Apresentados Nr. 22, 1985.
 [He3]
 , Some infinitedimensional MorseSmale systems defined by semilinear parabolic equations, J. Differential Equations 59 (1985), 165205. MR 804887 (86m:58080)
 [Hi]
 M. W. Hirsch, Differential topology, SpringerVerlag, New York, 1986. MR 1336822 (96c:57001)
 [Ma]
 H. Matano, Convergence of solutions of onedimensional semilinear parabolic equations, J. Fac. Sci. Univ. Tokyo 30 (1984), 221227. MR 501842 (80a:35016)
 [Po1]
 P. Poláčik, Convergence in smooth strongly monotone flows defined by semilinear parabolic equations, J. Differential Equations 79 (1989), 89110. MR 997611 (90f:58025)
 [Po2]
 , Complicated dynamics in scalar semilinear parabolic equations in higher space dimension, J. Differential Equations 89 (1991), 244271. MR 1091478 (92c:35063)
 [Si]
 L. P. Šilnikov, A contribution to the problem of the structure of an extended neighborhood of a structurally stable equilibrium of the saddlefocus type, Math. USSRSb. 10 (1970), 91102.
 [Ze]
 T. J. Zelenyak, Stabilization of solutions of boundary value problems for a second order parabolic equation with one space variable, Differential Equations 4 (1968), 1722 (translated from Differentsial'nye Uravneniya). MR 0223758 (36:6806)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199210894117
PII:
S 00029939(1992)10894117
Article copyright:
© Copyright 1992
American Mathematical Society
