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\begin{document}

\title{Partial Differential Equations}
\author{Homework 5}
\date{due October 30, 2002}
\maketitle

\begin{enumerate}
\item Prove a maximum principle for the following semilinear PDE, called
Burger's equation,
\begin{gather*}
  u_t + u \, u_x = u_{xx} \,, \\
  u(x,0) = g(x)
\end{gather*}
where $u=u(x,t)$ and $(x,t) \in \mathbb{R} \times [0,\infty)$.

\item Find the Fourier transforms for the following functions on
$\mathbb{R}$:
\begin{enumerate}
\item $f(x) = e^{-t |x|}$,
\item $\displaystyle f(x) =
\begin{cases}
1 & \text{for }|x| \leq 1 \\
0 & \text{for }|x| > 1
\end{cases}$.
\end{enumerate}
Use the result from (a) to solve the 1-dimensional differential
equation
\[
  u_{xx} - u = \delta \,,
\]
where $\delta$ is the Dirac measure centered at $x=0$.

\item Let $f,g \in L^1(\mathbb{R}^n)$, i.e.
\[
  \lVert f \rVert_{L^1}
  \equiv \int_{\mathbb{R}^n} |f(x)| \, dx < \infty \,;
\]
similarly for $g$.  Show
\begin{enumerate}
\item $\displaystyle \lim_{y\to 0}
  \int_{\mathbb{R}^n} |f(x) - f(x-y)| \, dx =0$.
\item $\displaystyle \lVert f*g \rVert_{L^1}
\leq \lVert f \rVert_{L^1} \, \lVert g \rVert_{L^1}$
\item Suppose that, moreover, $g \in L^\infty(\mathbb{R}^n)$.
Conclude that $f*g \in C(\mathbb{R}^n) \cap L^1(\mathbb{R}^n)$.

\end{enumerate}

\item Use the Fourier transform to re-derive the fundamental solution
of the heat equation.



\end{enumerate}


\end{document}
