Chaotic $C_0$-semigroups induced by semiflows in Lebesgue and Sobolev spaces

We give characterizations of chaos for $C_0$-semigroups induced by semiflows on $L^p_\rho(\Omega)$ for open $\Omega\subseteq\R$ similar to the characterizations of hypercyclicity and mixing of such $C_0$-semigroups proved in \cite{kalmes2009hypercyclic}. Moreover, we characterize hypercyclicity, mixing, and chaos for these classes of $C_0$-semigroups on $W^{1,p}_*(I)$ for a bounded interval $I\subset\R$ and prove that these $C_0$-semigroups are never hypercyclic on $W^{1,p}(I)$. We apply our results to concrete first order partial differential equations, such as the von Foerster-Lasota equation.