  We study the global approximate controllability of the reaction-diffusion
equation in a parallelpiped $ \Omega = (a_1,b_1 ) \times \ldots (a_n,b_n)
\subset R^n $, governed by a multiplicative control in a reaction term. It is
assumed that the initial state $ u_0 $ admits zeros only on the intersections
of $ \Omega$ with finitely many hyperplanes, parallel to the sides of $
\Omega$, and that $ u_0$ changes its sign after crossing such hyperplanes (we
further refer to them as the "hyperplanes of change of sign" or "zero
hyperplanes"). This paper can be viewed as a continuation of work presented in
\cite{CanKh, CanKh2} for the controllability of the one dimensional
reaction-diffusion equation with solutions admitting finitely many zeros.
However, the methods of \cite{CanKh, CanKh2} are intrinsically one dimensional,
while in this paper we introduce a novel approach to deal with the case of
multiple spatial variables.
