Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over ℝ d {\mathbb{R}^{d}} and in L p {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 I × ℝ d {I\times\mathbb{R}^{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.