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\begin{document}           


\centerline{{\sc Problem set 2}}
\centerline{Math 218A, Winter 2001}




\bigskip\noindent
{\bf 1.} 
Obtain the self-similar solution of
\beann
&&u_t = u_{xx},\\
&&u(x,0) = \left\{
\begin{array}{ll}
1 & \mbox{if $x<0$}, \\
0 & \mbox{if $x >0$}.
\end{array}\right.
\eeann




\bigskip\noindent
{\bf 2.} The porous medium equation in cylindrical coordinates is
\[
u_t = \frac{1}{r}\left(r\left(u^2\right)_r\right)_r.
\]
Look for similarity solutions of the form
\[
u(x,t) = \frac{1}{t^\alpha}\phi\left(\frac{r}{t^\beta}\right).
\]
Show that the choice $\alpha = 1/2$, $\beta=1/4$ is consistent with
the equation, and in that case derive a similarity 
solution which has compact support in $r$. (This solution, due
to Barenblatt, describes
the spread of a wetting front through the porous medium.)  

\bigskip\noindent
{\bf 3.} Suppose that $f \in C_c(\Rl^n)$ is compactly supported, 
and
\[
u\in C^2_1\left(\Rl^n\times (0,\infty)\right)
\cap C\left(\Rl^n\times [0,\infty)\right)
\]
is the bounded solution of
\beann
&&u_t = \Delta u,\\
&&u(x,0) = f(x).
\eeann
Prove that for $t>0$,
\beann
&&\|u(\cdot,t)\|_2 \le \|f\|_2,\\
&&\|u(\cdot,t\|_\infty \le \frac{1}{(4\pi t)^{n/2}} \|f\|_1,
\eeann
where $\|\cdot\|_p$ denotes the $L^p(\Rl^n)$-norm.


\bigskip\noindent
{\bf 4.} Let $\Omega$ be a smooth bounded domain. Suppose that
\beann
&&u_t + \mathbf{v}\cdot \nabla u = \Delta u \qquad\mbox{in $\Omega$},\\
&&\frac{\partial u}{\partial n} + \alpha u \ge 0
\qquad \mbox{on $\partial \Omega$},\\
&& u(x,0) \ge 0\qquad\mbox{in $\Omega$}
\eeann
where $\mathbf{v}$ is a constant vector, and $\alpha \ge 0$ is a nonnegative,
constant scalar. Assuming that the strong maximum principle
holds for this equation, prove that $u \ge 0$ for
$x\in \Omega$ and $t >0$. Give a physical interpretation in terms of heat flow.


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