Jacobson radicals of Ore extensions

Let $R$ be a ring, $\sigma$ be an automorphism of $R$, and $D$ be a $\sigma$-derivation on $R$. We will show that if $R$ is an algebra over a field of characteristic $0$ and $D$ is $q$-skew, then $J(R[x;\sigma,D])=I\cap R+I_0$ where $I=\{r\in R : rx\in J(R[x;\sigma,D])\}$ and $I_0=\{\sum_{i\geq 1}r_ix^i: r_i\in I\}$. We will prove that $J(R[x;\sigma,D])\cap R$ is nil if $\sigma$ is locally torsion and one of the following conditions is given: (1) $R$ is a PI-ring, (2) $R$ is an algebra over a field of characteristic $p>0$ and $D$ is a locally nilpotent derivation such that $\sigma D=D\sigma$. This answers partially an open question by Greenfeld, Smoktunowicz and Ziembowski.