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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

$\displaystyle {u_t} = \Delta u + f(x,u,\nabla u),\quad x \in \Omega ,\quad t > 0,$

on a smooth bounded convex domain $ \Omega \subset {\mathbb{R}^N}$ under Neumann boundary condition (2)

$\displaystyle \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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DOI: https://doi.org/10.1090/S0002-9939-1992-1089411-7
Article copyright: © Copyright 1992 American Mathematical Society

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