a. Consider the heat equation on $J=(-\infty,\infty)$ and prove that an energy
\begin{equation}
E(t)=\int_J u^2 (x,t)\,dx
\label{eq-HA3.5}
\end{equation}
does not increase; further, show that it really decreases unless $u(x,t)=\operatorname{const}$;
b. Consider the heat equation on $J=(0,l)$ with the Dirichlet or Neumann boundary conditions and prove that an $E(t)$ does not increase; further, show that it really decreases unless $u(x,t)=\operatorname{const}$;
c. Consider the heat equation on $J=(0,l)$ with the Robin boundary conditions
\begin{gather}
u_x(0,t)-a_0u(0,t)=0,\\[4pt]
u_x(l,t)+a_lu(l,t)=0.
\end{gather}
If $a_0>0$ and $a_l>0$, show that the endpoints contribute to the decrease of $E(t)=\int_0^l u^2 (x,t)\,dx$.
This is interpreted to mean that part of the energy is lost at the boundary, so we call the boundary conditions radiating or dissipative.
Hint. To prove decrease of $E(t)$ consider it derivative by $t$, replace $u_t$ by $ku_{xx}$ and integrate by parts.
Remark. In the case of heat (or diffusion) equation an *energy* given by (\ref{eq-HA3.5}) is rather mathematical artefact.
