  We consider the homogenization of a semilinear heat equation with vanishing
viscosity and with oscillating positive potential depending on $u/\varepsilon$.
According to the rate between the frequency of oscillations in the potential
and the vanishing factor in the viscosity, we obtain different regimes in the
limit evolution and we discuss the locally uniform convergence of the solutions
to the effective problem. The interesting feature of the model is that in the
strong diffusion regime the effective operator is discontinuous in the gradient
entry. We get a complete characterization of the limit solution in dimension
$n=1$, whereas in dimension $n>1$ we discuss the main properties of the
solutions to the effective problem selected at the limit and we prove
uniqueness for some classes of initial data.
