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\title{Introduction to Partial Differential Equations}
\author{Homework 5}
\date{due March 16, 2015}
\maketitle

\begin{enumerate}

\item Evans, p.~87 problem 12.
\item Evans, p.~87 problem 13.

\item Let $u(x,t)$ solve the heat equation
\begin{gather*}
  u_t - \Delta u = 0 \qquad \text{in } \R^n \times (0, \infty) \,, \\
  u = g \qquad \text{on } \R^n \times \{ t=0 \} 
\end{gather*}
with $g \in C(\R^n) \cap L^1(\R^n)$.  Show that
\[
  \| u \|_{L^\infty} \to 0 \qquad \text{as } t \to \infty
\]
while
\[
  \int_{\R^n} u(x,t) \, d x = \text{const} \,.
\]
Give a physical interpretation of each of the statements.

\item Find a solution formula for the heat equation with advection,
\begin{gather*}
  u_t - \Delta u + b \cdot D u = 0 \qquad 
  \text{in } \R^n \times (0, \infty) \,, \\
  u = g \qquad \text{on } \R^n \times \{ t=0 \} \,.
\end{gather*}
Hint: which equation is solved by $v(x,t)=u(x+bt,t)$?
\end{enumerate}

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