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{\bf  Math 241, Spring 2015\hfill Jerry L. Kazdan}
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\begin{center}
{\large\bf Problem Set 9}
\end{center}

{\sc Due:} Thurs. April 9 in class. [Late papers will be accepted until 1:00 PM Friday.]
\medskip

{\bf This week: } Please read Chapter 8 in the Haberman text. 
\bigskip

\begin{problems}

\item {}[See Section 7.3 in Haberman] Solve the wave equation
$u_{tt}=c^2\Delta u$ in the square $\Omega = \{0<x<\pi,\ 0<y<\pi\}$ in
the plane with $\nabla u\cdot N=0$ on the boundary and initial
conditions $u(x,y,0)=0$, \ $u_t(x,y,0) = \cos2x\cos5y$.

\item {}[See Section 7.3 in Haberman] In the square $0\le x \le a$, $0 \le y \le a$ in the plane, a substance is diffusing whose molecules multiply at a rate proportional to the concentration.  It thus satisfies
$$
u_t= k\Delta u + \gamma u,
$$
where $k$ and $\gamma$ are constants.  Assume that $u=0$ on all four sides of the square.  What is the condition on $\gamma$ so that the concentration does not grow without bound?

\item p.~287 \#7.4.1

\item p.~290 \#7.5.2

\item p.~291 \#7.5.6  \quad [This generalized Eq. 7.5.7 on page 289]

\item p.~338 \#7.10.1(a)

\item p.~339 \#7.10.2(b)

\end{problems}


\comments{
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\begin{center}{\large \bf Bonus Problem}
			
[Please give this directly to Professor Kazdan]
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\begin{problems}

\item [B-1] 
}


\vfill
[Last revised: \today]
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