Problem 4

[Vitaly Shemet]

Is initial Gaussian centered at $0$? Considering opposite I'm getting $M(T)$ and $m(T)$  neither increasing nor decreasing, what seems suspicious to me. If it is centered then $M(T)$ is decreasing... In other words, is $u(0,0)$ or $u(l,0) = max    u(x,t)$ for all $x$

[Victor Ivrii]

You need to find minima and maxima. Where they are located? -- you need to find this.

[Thomas Nutz]

I don't think that the maximum in the region $0\leq x \leq l$, $0 \leq t \leq T$ must either decreases or increase; I think it also can stay constant (e.g. iron rod that is initially very hot in the middle, then the maximum is found at t=0, x=l/2) and M(T) =const.

Am I wrong?

[Victor Ivrii]

I don't think that the maximum in the region $0\leq x \leq l$, $0 \leq t \leq T$ must either decreases or increase; I think it also can stay constant (e.g. iron rod that is initially very hot in the middle, then the maximum is found at t=0, x=l/2) and M(T) =const.

Am I wrong?

You are definitely correct. Since domain increases as $T,L$ grow, then maximum could only increase (or stay the same) and minimum could only decrease (or stay the same). The question is, what happens in the framework of the given problem

[Jinlong Fu]

q4

[Victor Ivrii]

Somehow this problem got shortened. Sure, $M(T)$ does not decrease as domain $\{0<x<l, 0<t<T\}$ increases and thus maximum over it can only increase or remain the same. Without boundary conditions we however cannot say anything more.

[Fanxun Zeng]

I just find by myself online for a well-typed clear solution for Problem 4 and share solution attached
http://www.math.uiuc.edu/~rdeville/teaching/442/hw2S.pdf

[Victor Ivrii]

I just find by myself online for a well-typed clear solution for Problem 4 and share solution attached
http://www.math.uiuc.edu/~rdeville/teaching/442/hw2S.pdf

Actually it contains unnecessary assumption about positivity of initial function $\phi$. Since $u=0$ on the boundary we know that $M(T)\ge 0$ and $m(T)\le 0$ anyway. Further maximum/minimum principle tells that $M(T)=\max \bigl(\max_{0\le x\le l} \phi(x),0\bigr)$ and  $m(T)=\min \bigl(\min_{0\le x\le l} \phi(x),0\bigr)$.