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\begin{document}

\title{Introduction to Partial Differential Equations}
\author{Homework 5}
\date{due May 15, 2017}
\maketitle

\begin{enumerate}

\item Evans, p.~290 problem 5.
\item Evans, p.~290 problem 8.
\item Evans, p.~290 problem 9.

\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 the
previous question 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 $u^\ini$.

\item In the setting of the Question 5, use the result from Question~2
to prove that
\[
  \limsup_{t \to \infty} \, \norm{u(t)}{\h1} \leq C   
\]
where an explicit estimate for $C$ can be given which, in particular,
shows that $C$ does not depend on the initial data $u^\ini$.  You may
assume that $u$ is sufficiently differentiable so that all your formal
manipulations are justified.



\end{enumerate}

\end{document}

