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I am reading the link http://math.mit.edu/~jspeck/18.152_Fall2011/Lecture notes/18152 lecture notes - 4.pdf

Says :

[tex]w_t-D w_{xx}=f [/tex] with f<0

w at [tex]\bar{Q}_T[/tex] has its maximum in [tex]\partial_p {Q}_T[/tex]. If w is strictly negative at [tex]\partial_p {Q}_T[/tex] then also is strictly negative in [tex]\bar{Q}_T[/tex]

(it is OK)

Says [tex]u=w-\epsilon t [/tex] , [tex]u \leq w [/tex], [tex]w \leq u + \epsilon T [/tex], T is cota,

then [tex]u_{t}-Du_{xx}=f-\epsilon <0 [/tex] (1)

(it is OK)

Says: Claim that the maximum of u in [tex]\bar{Q}_{T-\epsilon}[/tex] is on [tex]\partial_p {Q}_{T-\epsilon}[/tex]. To verify the claim we use [tex](t_0,x_0) \in \bar{Q}_{T-\epsilon}[/tex].

Says: [tex]t_0 \in (0,T-\epsilon][/tex] since if [tex]t=0[/tex] the claim is trueI dont understand this.

Says [tex]u_t=0[/tex] if [tex]t_0 \in (0,T-\epsilon)[/tex] (it is OK), but says [tex]u_t \geq 0[/tex] if [tex]t_0 =T-\epsilon[/tex]I dont understand this.

Then using Taylor and claims:

[tex]u_{t}-Du_{xx}>0 [/tex] (2) and says "which contradicts (1)"I dont understand this

Best regard.

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# I Weak maximum principle

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