If the solution to
$ u_t=u_{xx}+u_{yy}=0 u_t=u_{xx}+u_{yy} $
reaches an equilibrium solution, then $u_{t}=0$ at that equilibrium, so $u_{xx}+u_{yy}=0$. The author has shown that a necessary condition for $u$ to be minimizer of $E(u)$ is $u_{xx}+u_{yy}=0$.
This isn't steepest descent in the way that it is normally presented as an optimization algorithm for minimizing a function $f(x)$, but it is conceptually the same.
Of course you don't want to simply minimize $E(u)$ without respecting the original image- you want to somehow balance the minimization of $E(u)$ with keeping the original image. By starting the time dependent PDE with the original image and then stopping after a finite time, you can achieve this balance.
