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\title{Introductory Partial Differential Equations}
\author{Midterm Exam}
\date{March 21, 2013}
\maketitle

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\begin{enumerate}

\item Let $g = C^1(\R)$.  Solve the partial differential equation 
\begin{gather*}
  (x+t) (u_x + u_t) = 0  \qquad \text{in } \R \times (0,\infty) \,, \\
  u = g \qquad \text{on } \R \times \{ t= 0 \} \,.
\end{gather*}
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\item Suppose that a radial function $u=u(\lvert x \rvert)$ is
harmonic on $B(0,1) \subset \R^n$.  Show that $u \equiv \text{const}$.
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\item Suppose that $U \subset \R^n$ is open, connected, and bounded
with smooth boundary.  Suppose further that $u \in C^2(\overline U)$
solves the \emph{Neumann problem} for the Poisson equation
\begin{gather*}
  - \Delta u = f \qquad \text{in } U \,, \\
  \nu \cdot D u = g \qquad \text{on } \partial U
\end{gather*}
for some $f \in C(\overline U)$ and $g \in C(\partial U)$, where $\nu$
denotes the outer unit normal on $\partial U$.

Show that any other solution differs from $u$ by only a constant.
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\item Suppose that $u \in C_1^2(\R^n \times (0,\infty)) \cap C(\R^n
\times [0,\infty))$ solves the heat equation
\[
  u_t - \Delta u = 0 \qquad \text{in } \R^n \times (0,\infty)
\]
and is a Gaussian at the initial time, i.e., 
\[
  u(x,0) = a \, \e^{-b \, \lvert x \rvert^2}
\]
with some $a \in \R$ and $b>0$.  Prove that $u$ remains Gaussian for
all times $t>0$.
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\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 $C>0$ such that
\[
  \lvert Du(x,t) \rvert \leq \frac{C}{\sqrt t} \, \lVert g
  \rVert_{L^\infty} \,. 
\]
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\item Let $U \subset \R^n$ be open and bounded with smooth boundary.
Let $b \in C^1(\overline U)$ satisfy 
\begin{gather*}
  \operatorname{div} b \equiv D \cdot b = 0  \qquad \text{in } U \,, \\
  \nu \cdot b = 0 \qquad \text{on } \partial U \,.
\end{gather*}
Further, suppose that $u \in C^1(\overline U \times [0,T])$ solves the
transport equation
\[
  u_t + b \cdot D u = 0 \qquad \text{in } U \,.
\]

\begin{enumerate}
\item Show that
\[
  M = \int_U u \, dx
\]
is constant in time.

\item In a modeling scenario, $u$ could describe the concentration of
a certain substance in the container $U$.  Give a corresponding
physical interpretation of the result from (a).  Further, what is the
physical meaning of each of two conditions on $b$?

\end{enumerate}
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\end{enumerate}

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