  In this paper we study the global approximate multiplicative controllability
for nonlinear degenerate parabolic Cauchy problems. In particular, we consider
a one-dimensional semilinear degenerate reaction-diffusion equation in
divergence form governed via the coefficient of the \-reaction term (bilinear
or multiplicative control). The above one-dimensional equation is degenerate
since the diffusion coefficient is positive on the interior of the spatial
domain and vanishes at the boundary points. Furthermore, two different kinds of
degenerate diffusion coefficient are distinguished and studied in this paper:
the weakly degenerate case, that is, if the reciprocal of the diffusion
coefficient is summable, and the strongly degenerate case, that is, if that
reciprocal isn't summable. In our main result we show that the above systems
can be steered from an initial continuous state that admits a finite number of
points of sign change to a target state with the same number of changes of sign
in the same order. Our method uses a recent technique introduced for uniformly
parabolic equations employing the shifting of the points of sign change by
making use of a finite sequence of initial-value pure diffusion pro\-blems. Our
interest in degenerate reaction-diffusion equations is motivated by the study
of some \-energy balance models in climatology (see, e.g., the Budyko-Sellers
model) and some models in population genetics (see, e.g., the Fleming-Viot
model).
