  We study the Cauchy problem for the semilinear nonautonomous parabolic
equation $u_t=\mathcal{A}(t)u+\psi(t,u)$ in $[s,\tau]\times {{\mathbb R}^d}$,
$\tau> s $, in the spaces $C_b([s, \tau]\times{{\mathbb R}^d})$ and in $L^p((s,
\tau)\times{{\mathbb R}^d}, \nu)$. Here $\nu$ is a Borel measure defined via a
tight evolution system of measures for the evolution operator $G(t,s)$
associated to the family of time depending second order uniformly elliptic
operators $\mathcal{A}(t)$. Sufficient conditions for existence in the large
and stability of the null solution are also given in both $C_b$ and $L^p$
contexts. The novelty with respect to the literature is that the coefficients
of the operators $\mathcal{A}(t)$ are allowed to be unbounded.
