A simple characterization of chaos for weighted composition $C_0$-semigroups on Lebesgue and Sobolev spaces

TL;DR

This paper provides a simplified characterization of chaos for weighted composition $C_0$-semigroups on Lebesgue and Sobolev spaces, improving understanding of their dynamic behavior.

Contribution

It introduces a straightforward method to determine chaos in weighted composition $C_0$-semigroups on Lebesgue and Sobolev spaces, simplifying previous complex criteria.

Findings

Characterization of chaos on $L^p_ ho( ext{Ω})$ spaces.

Characterization of chaos on $W^{1,p}_*( ext{Ω})$ Sobolev subspaces.

Simplification of earlier chaos criteria for these semigroups.

Abstract

We give a simple characterization of chaos for weighted composition $C_0$-semigroups on $L^p_\rho(\Omega)$ for an open interval $\Omega\subseteq\mathbb{R}$. Moreover, we characterize chaos for these classes of $C_0$-semigroups on the closed subspace $W^{1,p}_*(\Omega)$ of the Sobolev space $W^{1,p}(\Omega)$ for a bounded interval $\Omega\subset\mathbb{R}$. These characterizations simplify previously obtained characterization of chaos for these classes of $C_0$-semigroups.