Math 6420 • PDE

\begin{enumerate}
\item[(a)] Derive the ``diffusion'' or ``heat'' equation in several spatial dimensions.
\item[(b)] In the derivation of this equation, it is routinely assumed that
\[
  \frac{d}{dt}\int_\Omega u(x,t)\,dx=\int_\Omega\frac{\partial u}{\partial t}\,dx.
\]
Under what conditions is this interchange of order valid? Prove your result. Hint: Examine
the function
\[
  w(t)=\int_0^t\left(\int_\Omega \frac{\partial u(x,t)}{\partial t}\,dx\right)\,dt.
\]
\end{enumerate}