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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

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: The scalar semilinear parabolic equation \[ {u_t} = \Delta u + f(x,u,\nabla u),\quad x \in \Omega ,\quad t > 0,\] on a smooth bounded convex domain $\Omega \subset {\mathbb {R}^N}$ under Neumann boundary condition (2) \[ \quad \frac {{\partial u}}{{\partial [unk]}} = 0\quad {\text {on }}\partial \Omega \] is considered. For any prescribed vector field $H$ on ${\mathbb {R}^N}$, a function $f$ is found such that the flow of (1), (2) has an invariant $N$-dimensional subspace and the vector field generating the flow of (1), (2) on this invariant subspace coincides, in appropriate coordinates, with $H$.


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Article copyright: © Copyright 1992 American Mathematical Society