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\begin{document}           


\centerline{{\sc Problem set 5}}
\centerline{Math 218A, Winter 2001}




\bigskip\noindent
{\bf 1.} Use Fourier transforms to solve the initial value
problem for the wave equation
\beann
&&u_{tt} - \Delta u = 0\\
&&u(x,0) = f(x),\\
&& u_t(x,0) = g(x).
\eeann
Show that for $f, g\in \S(\Rl^n)$ there is a unique solution
$u \in C^\infty(\Rl; \S)$. Show that
\[
\|u_t\|^2_{H^s} + \|\nabla u\|^2_{H^{s-1}}
\]
is constant. Let $D^s$ be the closure of $\S$ with respect to
the norm
\[
\|u\|_{D^s} = \|\nabla u\|^2_{H^{s-1}}.
\]
Deduce that there exists a unique generalized solution (defined as
a limit of Schwartz-valued solutions)
\[
u\in C(\Rl;D^s) \cap C^1(\Rl;H^{s-1})
\]
for each $f\in D^s$, $g\in H^{s-1}$.

\bigskip\noindent
{\bf 2.} Which of the following PDEs leads to a well-posed initial value
problem, with data given at $t=0$?
\beann
&&u_{tt} - u_{xx} = u,\\
&&u_{tt} = u_x,\\
&&u_{tt} = u_{xx}-u_{yy}.
\eeann
Remarks. The second equation is the ``sideways heat equation;'' with
$x$ and $t$ exchanged, the initial value problem corresponds to
the determination of the temperature inside a rod given the temperature
and heat flux at one end of the rod. The third equation is the wave equation 
with $x$ and $t$ exchanged; the initial value problem corresponds
to the imposition of Cauchy data on a  spacelike initial surface.

\bigskip\noindent
{\bf 3.} Use Fourier transforms to solve the system
\beann
&& u_t + v_x = 0,\\ 
&& v_t  = 0,\\
&&u(x,0) = f(x),\\
&& v(x,0) = g(x),
\eeann
for $f,g \in \S(\Rl)$. Show that the system satisfies the
Hadamard-Petrowsky condition. Show that the solution satisfies
the estimate
\[
\|(u,v)(t)|_{H^{s}} \le C \|(u,v)(0)|_{H^{s+1}}.
\]
(Note the ``loss'' of one derivative in this estimate;
as a result, the system is said to be weakly well-posed.)
Show that the system can be made ill-posed by the introduction
of a linear constant coefficient lower order term.


\bigskip\noindent
{\bf 4.} Solve the heat equation
\beann
&&u_t = u_{xx},\\
&&u(x,0) = f(x),
\eeann
by use of Laplace transforms in time.
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