Let consider on $[0,T]\times\mathbb{R}$ the heat equation:
\begin{equation} \tag{1} \label{heat} u_t = (\alpha u_x)_x\text{ with } u(0,x) = u_0(x)\in\mathcal{C}^\infty(\mathbb{R},\mathbb{R}). \end{equation}
where $\alpha:[0,T]\times\mathbb{R}\to\mathbb{R}$. I want to study the regularity of $u$ with $\alpha(t,\cdot)$ being piecewise constant.
First question
If $\varepsilon>0$ and $\alpha(t,x) = \varepsilon+\mathbf{1}_{x\geq0}$, is the solution of the problem (\ref{heat}) regular in space ?
Second question
If $\varepsilon>0$ and $\alpha(t,x) = \varepsilon+\mathbf{1}_{x\geq \lambda(t)}$ with $\lambda$ continuous, is the solution of the problem (\ref{heat}) regular in space ?
Any hint or references will be highly appreciated. Thank you very much!