  We study the Cauchy problem associated to a family of nonautonomous
semilinear equations in the space of bounded and continuous functions over R^d
and in L^p-spaces with respect to tight evolution systems of measures. Here,
the linear part of the equation is a nonautonomous second-order elliptic
operator with unbounded coefficients defined in IxR^d, (I being a
right-halfline). To the above Cauchy problem we associate a nonlinear evolution
operator, which we study in detail, proving some summability improving
properties. We also study the stability of the null solution to the Cauchy
problem.
