$-\triangle \phi + u \cdot \nabla \phi = e^{\phi}$ and $\phi \in C(\overline{\Omega})$ implies $\phi \in C^2(\Omega)$

Question

Suppose $U \subseteq \mathbb{R}^n$ is open, bounded, connected, $u$ is Lipschitz (or $C^1$ if it helps), and

$$-\triangle \phi + u \cdot \nabla\phi = e^{\phi}.$$ If I know $\phi \in C(\overline{\Omega})$ can I conclude at least $\phi \in C^2(\Omega)$?

This question is similar, except I don't know $u \in H^1(U)$.