Maximum Principle of the Diffusion Equation

Question

Consider a solution of the diffusion equation $u_{t} = u_{xx}$ in {$0\leq x\leq l,0\leq t<\infty$}

a) Let $ M(t)$ = the maximum of $u(x,t)$ in the closed rectangle {$0\leq x\leq l,0\leq t<T$}.Does $M(T)$ increase or decrease as a function of $T$?

So far my thinking is: if $T_{1} <T_{2}$

then the rectangle $R_{1}$ = {$0\leq x\leq l,0\leq t<T_{1}$} $\subset R_{2}$ = {$0\leq x\leq l,0\leq t<T_{2}$} and $M(T_{1}) < M(T_{2})$

Therefore as $T$ get larger so does $M(T)$ so its increasing but this contradictions two answers that I found

http://www.math.ku.edu/~slshao/fall2013math647Homework4.pdf http://www.math.uiuc.edu/~rdeville/teaching/442/hw2S.pdf

I want to know where my thinking is going wrong.

Answer 1

I just think two things One, that in the second reference they actually say M is constant (both decreasing and increasing) Second, if a set contains the other, then ok there is a relationship between the corresponding maximums, but not a strict one.