Heat Equation - Maximum Principle proof

1. May 16, 2010

epsilonjon

Hi,

i'm just going through my lecture notes reading a proof of the maximum principle for the heat equation. It goes roughly like this:

Maximum Principle:

Heat equation:

$$u_{t}=ku_{xx}$$
$$u(x,0)=\Psi(x)$$
$$x\in[0,L], t\in[0,T]$$.

Given a C2 solution of the HE then the maximum u(x,t) is attained at either t=0, x=0 or x=L.

Proof:

Define M=max{u(x,t)} on the set {t=0, x=0, x=L}. If we can show u(x,t)≤M on the entire rectangle then we are done.

Define $$v(x,t)=u(x,t)+ \epsilon x^{2} (\epsilon>0)$$. Want to show $$v(x,t)\leq M + \epsilon L^{2}$$, since then we would have

$$v(x,t) = u(x,t)+\epsilon x^{2} \leq M + \epsilon L^{2}$$
$$\Rightarrowu(x,t) \leq M + \epsilon(L^{2} - x^{2})$$

Taking $$\epsilon\rightarrow0$$ we then have $$u(x,t)\leqM$$.

Step 1: t=0, x=0, x=L

Obviously true.

Step 2: Interior

Claim v doesn't have a max on (0,L)x(0,T).

$$v_{t}-kv{xx}=(u+\epsilonx^{2})_{t} - k(u + \epsilonx^{2})_{xx} = u_{t} - k(u_{xx}+2\epsilon) = -2 \epsilon k < 0$$ (***)

If v(x,t) is a max on (0,L)x(0,T) then $$v_{t}(x,t)=0$$ and $$v_{xx}(x,t)\leq0$$. This would imply $$v_{t}(x,t)-kv_{xx}(x,t)\geq0$$. But we know $$v_{t}(x,t)-kv_{xx}(x,t)\leq0$$. Hence v doesn't have a max on (0,L)x(0,T).

Step 3: t=T

Now show v doesn't have a max on t=T.

If we freeze t=T then v(x,t) is a function of one variable:

$$v(x,t): (0,L)\rightarrow\Re$$.

We know

$$v_{x}(x,T)=0$$ and $$v_{xx}(x,T)\leq0$$ at the max. Assume (x,T) is the max of u on the top (t=T). Then $$v(x,T)-v(x, T-\delta) \geq 0$$ for small $$\delta>0$$. Therefore

$$v_{t}(x,T) = lim_{\delta\rightarrow0}[\frac{v(x,T)-v(x,T-\delta)}{\delta}] = lim_{\delta\rightarrow0}[\frac{v(x,T-\delta)-v(x,T)}{-\delta}] \geq 0$$.

So we have $$v_{t}(x,T) \geq 0 , v_{xx}(x,T) \leq 0$$ and so $$v_{t}-kv_{xx} \geq 0$$. But from (***) above we know that $$v_{t}-kv_{xx}<0$$. Hence v doesn't have a max on t=T.

But v is continuous on [0,L]x[0,T] so it must have a maximum there. Therefore the maximum must occur on either t=0, x=0 or x=L.

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Okay so that's the proof as we did it in lectures. I'm not really sure i'm understanding it that well though, because I don't see why you couldn't use the same argument as in step 3 to show that there cannot be a maximum on t=0 too?

Thanks for your help!

Last edited: May 16, 2010