I have to give a direct proof that if $U$ is bounded and $u ∈ C^2_1(U_T)∩C(\overline{U_T})$ solves the heat equation, then $$\max_{\overline{U_T}} u = \max_{Γ_T}u,$$ where $U_T := U ×(0,T]$ and $Γ_T := \overline{U_T} × \{0\} ∪(∂U ×[0,T])$. In the proof they claim: if $∆u−u_t ≥ 0$ then $$\max_{\overline{U_T}} u = \max_{Γ_T}u.$$
I'm not sure why is that enough to show.