Imbedding of any vector field in a scalar semilinear parabolic equation

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 $H$.