Toronto Math Forum
APM3462015S => APM346Home Assignments => HA3 => Topic started by: Victor Ivrii on February 05, 2015, 07:31:28 PM

Consider a solution of the diffusion equation $u_t=u_{xx}$ in $[0\le x \le l, 0\le t <\infty]$.
Let \begin{gather*}
M(T)= \max _{[0\le x \le l, 0\le t \le T]} u(x,t),\\[4pt]
m(T)= \min _{[0\le x \le l, 0\le t \le T]} u(x,t).
\end{gather*}
a. Does $M(T)$ increase or decrease as a function of $T$?
b. Does $m(T)$ increase or decrease as a function of $T$?

a.$M(T)= $max $u(x,t)$, where $0\leq x\leq l, 0\leq t\leq T]$
We claim that M(T) is an non decreasing function of T.
Let $T_1 \leq T_2$ and the rectangle $R_1 = {0\leq x\leq l,0\leq t \leq T_1} $ and $R_2 = {0\leq x\leq l,0\leq t \leq T_2} $.
The bottom and lateral sides of $R_1$ contained in $R_2$. By the uniqueness property, the diffusion equation of u on $R_2$ is an extension of u of $R_1$ and thus$M(T1)\leq M(T2)$
b.$m(T)=$ min $u(x,t)$, where $0\leq x\leq l, 0\leq t\leq T]$
We claim that m(T) is an non increasing function of T.
Let $T_1 \leq T_2$ and the rectangle $R_1 = {0\leq x\leq l,0\leq t \leq T_1} $ and $R_2 = {0\leq x\leq l,0\leq t \leq T_2} $.
The bottom and lateral sides of $R_1$ contained in $R_2$. By the uniqueness property, the diffusion equation of u on $R_2$ is an extension of u of $R_1$ and thus $m(T2)\leq m(T1)$

Actually it has nothing to do with maximum principle: since domain increases maximum can only increase (or remain the same) and minimum can only decrease (or remain the same).
Maximum/minimum principle would be useful to prove that both maximum/minimum remain the sameâ€”but then one needs boundary conditions.