**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.