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

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\newcommand*{\h}[2][]{\Space{H}{#1}{#2}}
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\newcommand*{\wl}[2][]{\Space{{\operatorname{w-}}L}{#1}{#2}}


\newcommand*{\ini}{{\operatorname{in}}}

\begin{document}

\title{Partial Differential Equations}
\author{Homework 6}
\date{due November 13, 2002}
\maketitle

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

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

Hint: This is similar to question 1 of the previous homework.

\item Let $\{u_n\} \subset \l2(U)$ be a weakly convergent sequence.
Show that $\{u_n\}$ is bounded.

\item
\begin{enumerate}

\item Prove the following elementary version of the \emph{Rellich
Theorem}:

\emph{The embedding $\Ht{t} \hookrightarrow \Ht{s}$ is compact for all real
numbers $s<t$.}

This is equivalent to saying that if $u_n \rightharpoonup u$ weakly in
$\Ht{t}$, then $u_n \to u$ strongly in $\Ht{s}$.

Hints: WLOG $u=0$; use the Fourier series representation of the
$\h{s}$ norms.

\item Use the result from part (a) to show that for every $T>0$ the
embedding
\[
  \l2([0,T]; \l2(\T)) \hookleftarrow
  C([0,T]; \wl2(\T)) \cap \wl2([0,T]; \wh1(\T)) \,,
\]
is compact, where the intersection on the right side is endowed with
the relative topology induced by the inclusion map.  In other words, a
sequence converges in the intersection iff it converges in each of the
spaces separately.

\end{enumerate}


\end{enumerate}


\end{document}
