Math 6410 • ODE

Let \(U\subset\R^n\) be an open bounded connected set with smooth boundary \(\partial U\).
Consider the initial boundary-value problem for the heat equation:
\[
\begin{cases}
  u_t-\Delta u=0, & \text{in }U\times(0,T],\\
  u=0, & \text{on }\partial U\times[0,T],\\
  u=g, & \text{on }U\times\{t=0\}.
\end{cases}
\]
\begin{enumerate}
\item[(a)] Use the mean value property for the heat equation
\[
  u(x,t)=\frac{1}{4r^n}\iint_{E(x,t;r)}
  u(y,s)\frac{\lvert x-y\rvert^2}{(t-s)^2}\dd y\dd s,
\]
where
\[
  E(x,t;r)=\{(y,s)\in\R^{n+1}:s\le t,\ \phi(x-y,t-s)\ge 1/r^n\}
\]
and \(\phi(x,t)\) is the fundamental solution of the heat equation, to prove the
\textit{strong maximum principle} for the heat equation.
\item[(b)] Show that the strong maximum principle implies the principle of
\textit{infinite propagation speed} for the problem above.
\end{enumerate}