2

The following is statement of weak maximum principle for heat equation. weak maximum princple

$(1)$ What does it mean mean by $C(\bar Q_T)$?
I mean I know $||f||_{C^k(\Omega)} = \sum_{n=0}^k \sup|f^{(n)}(x)|_{x \in \Omega}$ and order (or whatever it is called) of norm is not specified.

$(2)$ What does it mean by it attains it's maximum at boundary? Suppose I have a following Heat Equation . \begin{align*} u_t - u_{xx} = 0 & \; ; (t,x) \in (0,\infty)\times (0,1)) \\ u(x,0)=x(1-x) &\; ; x \in [0,1] \\ u(0, t) = u(1 , t)=0&\; ; t> 0 \end{align*}

The solution is given by $\displaystyle u(t,x) = \sum_{n=1}^\infty \left(\frac{2}{(2n-1)\pi} \right)^3 e^{-(2n-1)^2 \pi^2 t} \sin((2n-1)\pi x)$. It is clear that it at $t= 0$, it has maximum at $\{0\} \times \bar \Omega$ but still for $t>0$, it has maximum at the middle which is not on the boundary set $\partial \Omega$. What am I misunderstanding terribly?

I used Mathematica code

Animate[Plot[ Sum[(2/(Pi (2 n - 1)))^2 E^(-(2 n - 1)^2 Pi^2 t) Sin[(2 n - 1) Pi x], {n, 1, 50}], {x, 0, 1}, PlotRange -> {{0, 1}, {0, 1}}

and I got something like this enter image description here

Monkey D. Luffy
  • 1,632

1 Answers1

3

What does it mean mean by $C(\overline{Q_T})$

$C(K)$ is the same as $C^0(K)$. With upper index $0$, the definition of the norm includes only the zeroth derivative, i.e., function itself. This is the space of continuous functions on $\overline{Q_T}$ with the supremum norm.

Attains its maximum at boundary

To say that a function $w$, defined on a set $X$, attains its maximum on some set $A\subset X$, means that $\max_A w = \max_X w$. Or, in simpler terms, it means there is $a\in A$ such that $w(a)\ge w(x)$ for all $x\in X$.

In your computations you look at the maximum of $w$ for a fixed value of $t$. This is not what the theorem is about. It is about the maximum of $w$ as a function of two variables $x,t$ defined on the spacetime cylinder $\overline{Q_T}$. In your situation, the maximum of $w$ on $\overline{Q_T}$ is attained at $(x,t)=(1/2,0)$, which is a point on the boundary of $\overline{Q_T}$.

40 votes
  • 9,736