I've been asked to prove the following, but I don't find the way..
Let $\Omega\subset\left\{0<x_n<a\right\}$ be a subset of $\mathbb{R}^n$ such that it is bounded in the $n^{th}$ coordinate. Prove that the solution of the heat equation \begin{equation*} \begin{array}{lc} u_t=\Delta u && in\ \Omega \\ u=0 && on\ \partial\Omega \\ u(x,0)=u_0(x) && in\ \Omega \end{array} \end{equation*} satisfies \begin{equation*} \int_\Omega \dfrac{1}{2}u^2\leq M\cdot e^{-\delta t}, \end{equation*} for some $M,\delta>0$.
By showing this it is easily seen that any solution of this problem tends to $0$ as $t$ tends to $\infty$.
The problem that follows from this one is the next one:
Can one prove, with similar methods, that all the solutions of the problem \begin{equation*} \begin{array}{lc} u_t=\Delta u && in\ \Omega \\ \nabla u\cdot n=0 && on\ \partial\Omega \\ u(x,0)=u_0(x) && in\ \Omega \end{array} \end{equation*} where $n$ is the exterior normal vector of $\partial\Omega$, tend to steady solutions? Given an initial condition $u_0$, can one predict which is the value of the constant solution at which $u(x,t)$ tends as $t$ tends to $\infty$?
I'd be grateful for any help!
The operator $\Delta$ should be selfadjoint on the spatial domain with the desired boundary conditions. That allows you to write the solution as $u=e^{t\Delta}u_{0}$. The spectrum of $\Delta$ is normally going to be non-positive. So all of the modes will decay in time unless you have a non-trivial mode with eigenvalue $0$--that one can be present at $t=\infty$. In the first problem, that doesn't happen, and all of the modes die out.
In the second problem, the constant functions are eigenfunctions of the Laplacian operator because the domain consists of functions with the normal derivative equal to $0$. So the constant mode can persist without being damped in time. It fact, you know what the ultimate constant value will be: it will be the projection onto the eigenfunction which is the constant function. That projection is $$ \frac{(u_{0},1)}{(1,1)}1, $$ assuming the spatial region is finite. If the spatial region is finite, then the above is nothing more than the average of the initial date $u_{0}$ over the whole region, which is probably what you would expect for a heat solution inside an insulated body, which is what normal derivative $0$ means. Such a mode is unchanged throughout time.