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

We study the Cauchy problem associated to a family of nonautonomous semilinear equations in the space of bounded and continuous functions over ℝd and in Lp-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 × ℝd, (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. [ABSTRACT FROM AUTHOR]

Under suitable conditions, we obtain some characterizations of supercontractivity, ultraboundedness and ultracontractivity of the evolution operator G(t, s) associated to a class of nonautonomous second order parabolic equations with unbounded coefficients defined in I x ℝd, where I is a right-halfline. For this purpose, we establish an Harnack type estimate for G(t, s) and a family of logarithmic Sobolev inequalities with respect to the unique tight evolution system of measures {µt : t ∈ I} associated to G(t, s). Sufficient conditions for the supercontractivity, ultraboundedness and ultracontractivity to hold are also provided. [ABSTRACT FROM AUTHOR]

In this paper, we study the supercontractivity for Markov semigroups and obtain some sufficient and necessary conditions; especially explicit formulae are obtained for birth-death process and diffusion on the line. Sufficient conditions and necessary conditions in terms of isoperimetric inequalities are also presented. Moreover, we prove that the supercontractivity is equivalent to the compact embedding of Sobolev space into an Orlicz space. [ABSTRACT FROM AUTHOR]

This paper presents a self-contained account concerning a dimension-free Harnack inequality and its applications. This new type of inequality not only implies heat kernel bounds as the classical Li-Yau’s Harnack inequality did, but also provides a direct way to describe various dimension-free properties of finite and infinite-dimensional diffusion semigroups. The author starts with a standard weighted Laplace operator on a Riemannian manifold with curvature bounded from below, and then move further to the unbounded below curvature case and its infinite-dimensional settings. [ABSTRACT FROM AUTHOR]