Let $\varepsilon > 0$, consider the following viscous Hamilton-Jacobi equation $$ \begin{cases} u ^\varepsilon_t + H(x, Du ^\varepsilon) = \varepsilon\Delta u^\varepsilon& \text{in }\mathbb{R}^n \times (0,\infty) \\ u ^{\varepsilon} (x,0) = u_{0}(x) \text{on }\mathbb{R}^n \end{cases} $$
We assume that $u _{0}(x) \in C^2 (\mathbb{R}^n) \quad and \quad \| u _{0} \| _{C^2(\mathbb{R}^n)} < \infty$
$$ \begin{cases} H \in C^2 (\mathbb{R}^n\times\mathbb{R}^n)\\ H,D_{p}H \in BUC(\mathbb{R}^n\times B(0,R))\quad \text{for each } R > 0\\ \lim_{|p| \to \infty}\inf_{x \in \mathbb{R}^n} ((1/2) H(x,p)^2 + D_xH(x,p)\cdot p) = +\infty \end{cases} $$
**Question:**In my text book, it says that if we differentiate the equation, then we have $(u ^\varepsilon_t)_t + D_p H(x, Du ^\varepsilon)\cdot D u^\varepsilon_t = \varepsilon\Delta u^\varepsilon _t$
If we replace $u^\varepsilon _t = \varphi$,
then the equation $\varphi_t + D_p H(x, Du ^\varepsilon)\cdot D \varphi = \varepsilon\Delta \varphi$ is a linear parabolic equation. Thus, by the comparison principle for parabolic equations, we have for all $(x,t)\in \mathbb{R}^n\times[0,\infty)$, $\inf_{x \in \mathbb{R}^n}\varphi (x,0) \leq \varphi (x,t) \leq \sup_{x \in \mathbb{R}^n}\varphi (x,0)$. My question is what the comparison principle is used here, and does it use some assumption?