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\begin{document}

\title{Partial Differential Equations}
\author{Midterm Exam}
\date{October 29, 2008}
\maketitle

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\begin{enumerate}

\item 
Solve the partial differential equation
\begin{gather*}
  x \, u_t - t \, u_x = 0 \,,
\end{gather*}
where $u=u(x,t)$ for $(x,t) \in \R^2 \setminus (0,0)$.  What are the
characteristic curves?  What kind of initial/boundary data do you need
to prescribe?

\emph{Hint:} Start out with an ansatz for the charactistic curves of
the general form $x=x(s)$ and $t=t(s)$.
\hfill(10)

\item Show that if $u$ is harmonic on some open $U \subset \R^n$, then
  $u$ cannot have an isolated zero in $U$.
\hfill(5)


\item Let $U \in \R^n$ be open and bounded.  State a set of boundary
  conditions which are sufficient to guarantee that a solution $u \in
  C^4(\bar U)$ of the Poisson-type problem for the bi-Laplacian,
\[
  \Delta^2 u = f \,,
\] 
satisfying those boundary conditions, is unique.

\emph{Hint:} Energy methods.
\hfill(10)


\item Consider the following initial-boundary value problem (IBVP) for the
  heat equation on $U=(-\pi/2,\pi/2)$,
\begin{align*}
  u_t - u_{xx} = 0 \qquad &\text{in } U \times (0, \infty) \,,
  \\
  u = 0 \qquad &\text{on } \{x = \pm \pi/2 \} \times (0,\infty) \,, \\   
  u = g \qquad &\text{on } U \times \{ t = 0 \} \,.
\end{align*}
\begin{enumerate}
\item Let $u_i \in C_1^2(U \times (0, \infty))$
  solve the IBVP with initial data $g_i \in C(\bar U)$ for
  $i=1,2$.  Show that if $g_1\leq g_2$, then $u_1 \leq u_2$ for all
  $(x,t) \in \bar U \times [0,\infty)$.

\emph{Note:} You may quote a well-known theorem from class; no need to
  prove from scratch.

\item Show that the IBVP has a particular solutions of the form
\[
  u(x,t) = a(t) \, \cos x \,.
\]
Derive an expression for $a(t)$.

\item Conclude that 
\[
  \lvert u(x,t) \rvert \leq c \, \mathrm{e}^{-t}
\]
where $c$ depends only on $g$.

\emph{Note:} For simplicity of the argument, you may assume that $g$
is compactly supported in $U$.
\end{enumerate}
\hfill(5+5+5)

\item Let $f \in C^2(\R)$ and $b \in \R^n$.  Show that $f(b \cdot x -
  t)$ solves the wave equation in $\R^n$.  Describe the geometry of
  the solution.  What is the speed of propagation?
\hfill(10)

\end{enumerate}

\end{document}
