i have a question about strong maximum principle and mean value formula for heat equuation i saw two versions of these theorems one (In Evans Book) with hypothesis that $u \in C^{2,1}(U\times (0,T])$
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howevever in other pdf i found that the same theorems holds true if instead i
assume that $u \in C^{2,1}(U\times [0,T))$
Can i say that these theorems are true if i assume that $U_{T}=U\times (0,T)$ i mean that $u \in C^{2,1}(U\times (0,T))$ and $(x_{0},t_{0})$ is in $U_{T}=U\times (0,T)$ for strong maximum principle ?
my second question can one prove that the set
$$K={(x,t)\in U\times (0,T)):u(x,t)=u(x_{0}, t_{0})}$$
is closed(u is continue ) and open because of mean value formula ?here $(x_{0}, t_{0}) \in U_{T}=U\times (0,T)$ such that :
$$u(x_{0}, t_{0})=\max_{\overline{U}}u(x,t) $$
and conclude because U is a connected is this proof also true ?
Thanks
