\documentclass[12pt]{article}
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\newcommand*  {\N} {{\mathbb N}}
\newcommand*  {\R} {{\mathbb R}}

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\newcommand*{\bnorm}[3][]{\bigl\lVert #2 \bigr\rVert_{#3}^{#1}}
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\newcommand*{\h}[2][]{\Space{H}{#1}{#2}}
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\newcommand*{\ini}{{\operatorname{in}}}

\begin{document}

\title{Partial Differential Equations}
\author{Homework 8}
\date{due April 25, 2007}
\maketitle

In the following, $\T$ denotes the 1-torus, i.e.\ $\T = \R \,
\operatorname{mod} 2\pi$.

\begin{enumerate}

\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 $\displaystyle \lim_{y\to 0}
  \int_{\mathbb{R}^n} |f(x) - f(x-y)| \, dx =0$.

\emph{Hint:} Use mollifiers.

\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
\begin{enumerate}
\item Consider a sequence $u_n \in \l2$ with $u_n \rightharpoonup u
\in \l2$ weakly.  Show that
\begin{equation*}
  \norm{u}{} \leq \liminf_{n\to\infty} \norm{u_n}{} \,.
  \tag{*}
\end{equation*}
(Remark: This statement is actually true for any Banach space.)

\item Give an example where (*) holds with strict inequality.

\end{enumerate}

\item Prove that $C(\T) \supset \Ht1$.

\item
\begin{enumerate}
\item Show that, for every $u \in \Lt{r}$ with $2\leq r<\infty$,
\[
  \norm{u}{\l2} \leq (2\pi)^{\tfrac{r-2}{2r}} \, \norm{u}{\l{r}} \,. 
\]
\emph{Hint:} H\"older inequality.

\item Consider the Fisher--Kolmogorov equation on $\T$,
\begin{gather*}
  u_t = u_{xx} + (1-u) \, u^m \,, \\
  u(0) = u^\ini \,,
\end{gather*}
where $m$ is an even positive integer.  Use the result from (a) to
sharpen the $\l2$ estimate derived in the lecture as follows:  Show
that
\[
  \limsup_{t \to \infty} \, \norm{u(t)}{\l2} \leq C   
\]
where an explicit estimate for $C$ can be given which, in particular,
shows that $C$ does not depend on the initial data $u^\ini$.

\end{enumerate}

\item Show that if $u^\ini \geq 0$, the solution $u(t)$ to the
Fisher--Kolmogorov equation remains nonnegative for every $t \geq 0$.
You may assume that $u$ is as smooth as you need.

\emph{Hint:} This is similar to Homework~6, Question~2.

\end{enumerate}

\end{document}
