Hypercontractivity, supercontractivity, ultraboundedness and stability in semilinear problems

TL;DR

This paper investigates the properties of solutions to nonautonomous semilinear PDEs, focusing on hypercontractivity, supercontractivity, ultraboundedness, and stability, using evolution systems of measures and nonlinear evolution operators.

Contribution

It introduces a detailed analysis of summability improving properties and stability for nonautonomous semilinear equations with unbounded coefficients.

Findings

Proves summability improving properties of the nonlinear evolution operator.

Establishes stability criteria for the null solution.

Analyzes behavior in both bounded continuous functions and L^p-spaces.

Abstract

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.