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Is there a maximum principle for the heat equation

$\partial_t u(x,t)=k \partial_{xx}^2 u(x,t)$

for $(x,t)\in[O,L] \times [0, \infty]$?

If $u$ has a maximum it would occur at $t=0$, $x=0$ or $x=L$, just like in the bounded case, but since we can't assume $u$ attains its maximum, I don't know whether it has a maximum principle or how to prove it.

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I'm not assuming any boundary conditions since you didn't specify any.. but an analogous argument will work if there are.

If you separate variables, you get that $u(x,t)$ is of the form $$u(x,t) = \sum_{n=0}^{\infty} A_n \cos({n\pi \over L}x)e^{-{n^2\pi^2k^2 \over L^2}t}+ \sum_{n=1}^{\infty} B_n \sin({n\pi \over L}x)e^{-{n^2\pi^2k^2 \over L^2}t} $$ Notice that as $t$ goes to infinity, all terms go to zero except the first term of the cosine series. In fact as $t \rightarrow \infty$, $u(x,t)$ converges uniformly to $A_0$, which is the initial average temperature from $x = 0$ to $x = L$. Unless $u(x,0)$ is constant (in which case $u(x,t)$ is constant and there is nothing to prove), there is going to be some $x_0$ for which $|u(x_0,0)| = A_0 > A$.

You can prove a maximal principle as follows. Let $t_0$ be such that $|u(x,t)| < A_0$ for all $t \geq t_0$. Then by the maximal principle on the box $[0,L] \times [0,t_0]$, $|u(x,t)|$ achieves its maximum somewhere on the boundary of the box. On $[0,L] \times [t_0,\infty)$, $|u(x,t)|$, being less than $A_0$, is less than $|u(x_0,0)|$, which is in turn at most the maximum on the boundary of the box.

So $|u(x,t)|$ achieves its overall maximum on the boundary of the box $[0,L] \times [0,t_0]$. It can't occur on the side $[0,L] \times \{t_0\}$ because $|u(x,t)| < A_0$ there. Hence it achieves its maximum on one of the other three sides, which are all part of the boundary of the original domain $[0,L] \times [0,\infty)$. So we conclude that $|u(x,t)|$ does achieve its supremum over $[0,L] \times [0,\infty)$, and on the boundary of that domain.

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Actually it is true for general second order parabolic equations of the form

$$ \displaystyle \partial_t u = \sum_{i,j=1}^{n}a_{ij}(x,t)\partial_i\partial_ju+\sum_{k=1}^{n}b_k(x,t)\partial_k u + c(x,t) u $$

on a bounded spatial domain $\Omega\subset\mathbb{R}^n$ and time interval $t\in(0,\infty)$, where the coefficients $a,b,c$ are continuous on $\bar\Omega\times[0,\infty)$ and $[a_{ij}]$ is symmetric positive definite. With some growth condition at infinity it works for unbounded spatial domains as well.

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