Modification of maximum principle for sub solution of heat equation.

This is a question from Evans . I know the well known maximum principle for heat equation. What would be the modification in the following case ? Any explanation will be appreciated. Thanks for your help.

We say that $v\in C_{1}^{2}(U_{T})$ is a subsolution of the heat equation if: $$v_{t}-\Delta v\leq 0\qquad\text{in }U_{T}$$ Prove for a subsolution $v$ that: $$v(x,t)\leq\frac{1}{4r^{n}}\iint_{E(x,t;r)}{v(y,s)\frac{|x-y|^{2}}{(t-s)^{2}}\:dy\:ds}$$ For all $E(x,t;r)\subseteq U_{T}$. Prove that therefore: $$\max_{\overline{U}_{T}}{v}=\max_{\Gamma_{T}}{v}$$

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