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\title{Introductory Partial Differential Equations}
\author{Midterm Exam}
\date{March 23, 2015}
\maketitle

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\begin{enumerate}

\item Solve the inhomogeneous transport equation
\begin{gather*}
  u_t + u_x + t \, \mathrm{e}^{-x} = 0 
  \qquad \text{in } \R \times (0, \infty) \,, \\
  u(x,0) = \mathrm{e}^{-x} \qquad \text{on } \R \,.
\end{gather*}
\hfill(10)


\item Suppose $u \colon \R^n \to \R$ is harmonic with
\[
  \int_{\R^n} \lvert u \rvert \, d x < \infty \,.
\]
Show that this implies $u=0$.
\hfill(5)


\item Suppose $u$ is a harmonic function on $\R^n$ with $n\geq 2$ such
that 
\[
  u(x)=x_1+x_2 \qquad \text{on } \partial B(0,1) \,.
\]
\begin{enumerate}
\item What are the minimum and maximum values of $u$ on
$\overline{B(0,1)}$?
\item Find $u(0)$.
\end{enumerate}
\hfill(5+5)



\item Let $U \subset \R^n$ be open and bounded with smooth boundary.
Let $u \in C^2(\overline U)$ be a solution to the \emph{Helmholtz
equation} with Neumann boundary conditions,
\begin{gather*}
  -\Delta u + u = f \qquad \text{in } U \,, \\
  \nu \cdot D u = g \qquad \text{on } \partial U \,.
\end{gather*}
\begin{enumerate}
\item Show that $u$ is the unique such solution.
\item What can you say about uniqueness of solutions to the Poisson
equation $\Delta u = f$ in otherwise the same setting?
\end{enumerate}
\hfill(5+5)


\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 $t>0$,
\[
  \Phi(z,t) = \frac1{\sqrt{4\pi t}} \,
              \mathrm e^{-\tfrac{\lvert z \rvert^2}{4t}} \,.
\]

\begin{enumerate}
\item Show that for every $g \in L^1(\R)$ there exists a constant
$c>0$ such that
\[
  \sup_{x \in \R} \, \lvert u(x,t) \rvert \leq \frac{c}{\sqrt t} \,.
\]

\item Show that for every $g = h_x$ with $h \in L^1(\R)$ there exists
a constant $c>0$ such that
\[
  \sup_{x \in \R} \, \lvert u(x,t) \rvert \leq \frac{c}{t} \,.
\]

\item Give a qualitative explanation for (b) vs.\ (a).
\end{enumerate}
\hfill(5+5+5)


\item Consider the equation\footnote{This equation is most commonly
known as the Camassa--Holm equation, though its derivation goes back
to Fokas and Fuchssteiner.}
\begin{gather*}
  u_t-u_{xxt}+3\,u\,u_x=2\,u_x\,u_{xx}+u\,u_{xxx} 
  \qquad \text{in } \R \times (0,\infty) \,, \\
  u = g \qquad \text{on } \R \times \{ t=0 \} \,.
\end{gather*}
Suppose that $u$ is a solution such that $u(x,t) \to 0$ as $x \to
\pm\infty$ with $u_x$ and $u_{xx}$ bounded for every fixed $t \geq 0$.

Show that
\[
  M(t) = \int_\R \bigl( u^2 + u_x^2 \bigr) \, dx
\]
is a constant of the motion.
\hfill(10)


\end{enumerate}

\end{document}
