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\begin{document}

\title{Partial Differential Equations}
\author{Midterm Exam}
\date{October 28, 2004}
\maketitle

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\begin{enumerate}

\item Solve the partial differential equation
\begin{gather*}
  u_t + x \, u_x = 0
  \quad \text{in } \mathbb{R} \times (0,\infty) \,, \\
  u = g
  \quad \text{on } \mathbb{R} \times \{t=0\} \,,
\end{gather*}
where $u=u(x,t)$ and $g \in \Space{C}{}{1}(\mathbb{R})$.

\emph{Hint:} Show that $z(s)=u(x(s),t+s)$ is constant if
$x'(s)=x(s)$.
\hfill (10)

\item For $U \subset \mathbb{R}^n$ open and (path-)connected, let $u
\in \Space{C}{}{2}(U)$ be harmonic with $u \geq 0$.

Show that if $u(x)>0$ for some $x \in U$, then $u>0$ everywhere in
$U$.
\hfill (10)

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

Assume that $g$ is continuous and compactly supported.  Show
that there exists a $C>0$, depending only on the support of $g$, such
that
\[
  \lvert Du(x,t) \rvert \leq \frac{C}{\sqrt t} \, \lVert g
  \rVert_{L^\infty} \,. 
\]

\hfill (10)

\item Let $U \subset \mathbb{R}^n$ be open and bounded with
$\Space{C}{}{1}$ boundary.  Assume that $u \in \Space{C}{1}{2}(\bar
U\times[0,\infty))$ solves 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\} \,, \\
  u = 0
  \quad \text{on } \partial U \times (0,\infty) \,, \\
\end{gather*}
and define the ``energy''
\[
  E(t) = \int_U \lvert u(x,t) \rvert^2 \, dx \,.
\]

Prove that $E(t) \leq E(0)$ for every $t \geq 0$.
\hfill (10)

\item Let $h \in \Space{C}{}{2}(\mathbb{R}^3)$ and set
\[
  u(x,t) = \PVint_{\partial B(x,t)} t \, h(y) \, dS(y)
\]
for $t>0$.  

Show that
\begin{enumerate}
\item $\displaystyle \lim_{t\to 0} u(x,t) = 0$.
\item $\displaystyle \lim_{t\to 0} u_t(x,t) = h(x)$.
\item \textbf{(Extra credit.)} $u$ solves the wave equation
\[
  u_{tt} - \Delta u = 0
\]
on $\mathbb{R}^3 \times (0,\infty)$.
\end{enumerate}
\hfill (5+5\textit{+5})

\end{enumerate}

\end{document}
