## Heat Equation

Let $\sigma$ denote a solution to the heat equation on [0,1] under the Dirichlet boundary conditions of $f$(0)= $a$ and $f$(1)= $b$ and IVP of $\sigma$( $x$,0)= $f$( $x$), and let $u$ denote the corresponding steady state solution. Show that the lim as t approaches infinity of $\sigma$( $x$, $t$)= $u$( $x$).