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\def\R{\mathbb{R}}

\begin{document}

\title{Partial Differential Equations}
\author{Midterm Exam}
\date{April 11, 2007}
\maketitle

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\begin{enumerate}

\item 
\begin{enumerate}
\item
Solve the partial differential equation
\begin{gather*}
  u_t + x^2 \, u_x = 0 \,,
\end{gather*}
where $u=u(x,t)$ on the open first quadrant of the $(x,t)$ plane.

\emph{Hint:} Show that $z(s)=u(x(s),t+s)$ is constant if
$x'(s)=x^2(s)$.
\item Draw the characteristic curves, then state a set of boundary
and/or initial conditions that specify the solution uniquely in the
first quadrant of the $(x,t)$ plane.
\end{enumerate}
\hfill (10+10)

\item Let $U \subset \R^n$ open and bounded and $u \in C^2(\bar
U)$.  We say that $u$ is \emph{superharmonic} if
\[
  -\Delta u \geq 0 \quad \text{in } U \,.
\] 

\begin{enumerate}
\item Prove that for superharmonic $u$,
\[
  u(x) \geq \PVint_{B(x,r)} u(y) \, dy
\]
for all $B(x,r) \subset U$.

\emph{Note:} Do the computation explicitly.  Simply referring to the
homework is not enough.

\item Show that if $u$ is superharmonic and $u\geq 0$ on $\partial U$,
then $u \geq 0$ in $\bar U$.
\end{enumerate}
\hfill (10+10)

\item Recall that the solution to the heat equation
\begin{gather*}
  u_t - \Delta u = 0
  \quad \text{in } \mathbb{R} \times (0,\infty) \,, \\
  u = g
  \quad \text{on } \mathbb{R} \times \{t=0\} 
\end{gather*}
is given by
\[
  u(x,t) = \int_{\mathbb{R}} \Phi(x-y,t) \, g(y) \, dy \,,
\]
where, for $t>0$,
\[
  \Phi(z,t) = \frac1{\sqrt{4\pi t}} \,
              e^{-\tfrac{\lvert z \rvert^2}{4t}} \,.
\]

\begin{enumerate}
\item Show that if $g$ is an even function, i.e.\ if $g(x)=g(-x)$, then
$u(\cdot,t)$ is even for every $t\geq 0$.
\item What does this imply for the solution of the heat equation on
the halfline with Neumann boundary conditions, i.e.
\begin{gather*}
  u_t - \Delta u = 0
  \quad \text{in } (0,\infty) \times (0,\infty) \,, \\
  u_x = 0 
  \quad \text{on } \{x=0\} \times (0,\infty) \,, \\
  u = g
  \quad \text{on } (0,\infty) \times \{t=0\} \,?
\end{gather*}
\end{enumerate}
\hfill (10+10)

\item Let $U \subset \R^n$ open and bounded.  Recall that for $1 \leq
p < \infty$, the $\Space{L}{}{p}$-norm of a suitably integrable
function $u$ is defined
\[
  \lVert u \rVert_{\Space{L}{}{p}}^p
  = \int_U \lvert u(x) \rvert^p \, dx
\]
while the $\Space{L}{}{\infty}$-norm is given by
\[
  \lVert u \rVert_{\Space{L}{}{\infty}}
  = \operatorname*{ess\,sup}_{x\in U} \, \lvert u(x) \rvert \,.
\]
\begin{enumerate}
\item Show that if $u \in \Space{C}{1}{2}(\bar
U\times[0,\infty))$ satisfies the heat equation
\begin{gather*}
  u_t - \Delta u = 0
  \quad \text{in } U \times (0,\infty) \,, \\  
  u = g
  \quad \text{on } U \times \{t=0\} \,, \\
  \nu \cdot D u = 0
  \quad \text{on } \partial U \times (0,\infty) \,, \\
\end{gather*}
then
\[
  \lVert u (\cdot, t) \rVert_{\Space{L}{}{p}} \leq 
  \lVert g \rVert_{\Space{L}{}{p}} \,,
\]
for $p\geq 2$.

\emph{Note:} For simplicity, you may assume that $p$ is an even
integer.

\item Prove that, provided $u \in C(\bar U)$,
\[
  \lim_{p \to \infty} 
  \lVert u  \rVert_{\Space{L}{}{p}}
  = \lVert u  \rVert_{\Space{L}{}{\infty}} \,.
\]

\emph{Note:} The statement is also true if $u$ is only
$\Space{L}{}{\infty}$, but the proof is not so elementary.

\item Explain why (a) and (b) imply yet another proof of the maximum
principle for the heat equation.

\end{enumerate}
\hfill (10+10+10)

\item Show that for solutions of the wave equation
\[
  u_{tt} - \Delta u = 0
\]
on $\mathbb{R}^n \times (0,\infty)$, where $u(\,\cdot\,,0)$ has
compact support, the energy
\[
  E(t) = \int_{\R^n} \bigl( u_t^2 + \lvert Du \rvert ^2 \bigr) \, dx 
\]
is constant in time.
\hfill (10)

\end{enumerate}

\end{document}
