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We study the global approximate controllability of the reaction-diffusionequation 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 isassumed that the initial state $ u_0 $ admits zeros only on the intersectionsof $ \Omega$ with finitely many hyperplanes, parallel to the sides of $\Omega$, and that $ u_0$ changes its sign after crossing such hyperplanes (wefurther refer to them as the "hyperplanes of change of sign" or "zerohyperplanes"). This paper can be viewed as a continuation of work presented in\cite{CanKh, CanKh2} for the controllability of the one dimensionalreaction-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 ofmultiple spatial variables.

Multiplicative controllability of the reaction-diffusion equation on a parallelepiped with finitely many zero hyperplanes