Imbedding of any vector field in a scalar semilinear parabolic equation

Author:
P. Poláčik

Journal:
Proc. Amer. Math. Soc. **115** (1992), 1001-1008

MSC:
Primary 35K60

DOI:
https://doi.org/10.1090/S0002-9939-1992-1089411-7

MathSciNet review:
1089411

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Abstract | References | Similar Articles | Additional Information

Abstract: The scalar semilinear parabolic equation

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 .

**[Am]**H. Amann,*Existence and regularity for semilinear parabolic evolution equations*, Scuola Norm. Sup. Pisa Cl. Sci. (4)**11**(1984), 593-696. MR**808425 (87h:34088)****[An]**S. B. Angenent,*The Morse-Smale property for a semilinear parabolic equation*, J. Differential Equations**62**(1986), 427-442. MR**837763 (87e:58115)****[F-M]**B. Fiedler and J. Mallet-Paret,*A Poincaré-Bèndixson theorem for scalar reaction diffusion equations*, Arch. Rational Mech. Anal.**107**(1989), 325-345. MR**1004714 (90j:35116)****[F-P]**B. Fielder and P. Poláčik,*Complicated dynamics of scalar reaction diffusion equations with a nonlocal term*, Proc. Roy. Soc. Edinburgh**115A**(1990), 167-192. MR**1059652 (91g:45008)****[G-T]**D. Gilbarg and Trudinger,*Elliptic partial differential equations of second order*, Springer-Verlag, Berlin, 1977. MR**0473443 (57:13109)****[G-H]**J. Guckenheimer and P. Holmes,*Nonlinear oscillations, dynamical systems, and bifurcations of vector fields*, Springer-Verlag, New York, 1983. MR**709768 (85f:58002)****[He1]**D. Henry,*Geometric theory of semilinear parabolic equations*, Lecture Notes in Math., vol. 840, Springer-Verlag, 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 infinite-dimensional Morse-Smale systems defined by semilinear parabolic equations*, J. Differential Equations**59**(1985), 165-205. MR**804887 (86m:58080)****[Hi]**M. W. Hirsch,*Differential topology*, Springer-Verlag, New York, 1986. MR**1336822 (96c:57001)****[Ma]**H. Matano,*Convergence of solutions of one-dimensional semilinear parabolic equations*, J. Fac. Sci. Univ. Tokyo**30**(1984), 221-227. MR**501842 (80a:35016)****[Po1]**P. Poláčik,*Convergence in smooth strongly monotone flows defined by semilinear parabolic equations*, J. Differential Equations**79**(1989), 89-110. MR**997611 (90f:58025)****[Po2]**-,*Complicated dynamics in scalar semilinear parabolic equations in higher space dimension*, J. Differential Equations**89**(1991), 244-271. 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 saddle-focus type*, Math. USSR-Sb.**10**(1970), 91-102.**[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), 17-22 (translated from Differentsial'nye Uravneniya). MR**0223758 (36:6806)**

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DOI:
https://doi.org/10.1090/S0002-9939-1992-1089411-7

Article copyright:
© Copyright 1992
American Mathematical Society