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? I know something like this about heat equation. Please teach me.
2026-03-25 23:44:03.1774482243
Maximum principle or comparison principle
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