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\newcommand\PVint{\mathop{\setbox0\hbox{$\displaystyle\intop$}%
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\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Homework 6}
\date{due April 8, 2015}
\maketitle

\begin{enumerate}
\item Evans, p.~88 Problem 17.
\item For $h \in C^2(\mathbb{R}^3)$ with compact support and define,
for $t>0$,
\[
  u(x,t) = \PVint_{\partial B(x,t)} t \, h(y) \, dS(y) \,.
\]
Show that
\begin{enumerate}
\item There exists $C>0$ such that $\lvert u(x,t) \rvert \leq C/t$ for
all $t>0$.
\item $\displaystyle \lim_{t\to 0} u(x,t) = 0$.
\item $\displaystyle \lim_{t\to 0} u_t(x,t) = h(x)$.
\item $u$ solves the wave equation
\[
  u_{tt} - \Delta u = 0
\]
on $\mathbb{R}^3 \times (0,\infty)$.
\end{enumerate}


\item Evans, p.~163 Problem 3.
\item (This question refers to the remark in Evans, p.~114.)  Consider
the scalar conservation law in one spatial dimension
\begin{gather*}
  u_t + F'(u) \, u_x = 0 
  \qquad \text{in } \mathbb R \times (0,\infty) \,, \\
  u = g \qquad \text{in } \mathbb R \times \{ t = 0 \} \,.
\end{gather*}
\begin{enumerate}
\item Show that the function $u$ defined through the implicit equation
\[
  u = g(x-t \, F'(u))
\]
is a solution provided
\begin{equation*}
  1 + t \, g'(x-t \, F'(u)) \, F''(u) \neq 0 \,.
  \tag{*}
\end{equation*}
\item 
Suppose that $F(u) = \tfrac12 \, u^2$.  When does solvability
condition (*) fail?  Is this only a failure of the method of
characteristics, or does it also correspond to a failure of the
solution to the PDE?
\end{enumerate}

\end{enumerate}
\end{document}

