08 - The Maximum Principle for the Heat Equation

The Maximum Principle for the Heat Equation

Suppose $u(x,t)$ satisfies

$$u_t-u_{xx}\le 0$$

in a region $D_{\tau }$ bounded by

1. lines $t=0$ and $t=\tau >0$
2. two non-intersecting smooth curves $C_1$ and $C_2$ which are nowhere parallel to $x-$axis

Suppose also that $f\le 0$ in $D_{\tau }$ then

$u$ takes maximum value either on $t=0$ or on one of the curves $C_1$ or $C_2$

If $u_t-u_{xx}\ge 0$ then $u$ attains minimum value on $\partial D_{\tau }$ but not on $t=tau$

Proof