Math 6410 • ODE

Consider the problem
\[
\begin{aligned}
  u_t &= u_{xx}, && t\in(0,1),\ x\in(0,1),\\
  u(0,t)=u(1,t) &= 0, && t\in(0,1),\\
  u(x,1) &= f, && x\in(0,1).
\end{aligned}
\]
\begin{enumerate}
\item[(a)] What are the least restrictive condition you can find on \(f\) which guarantee
that the problem admits a solution \(u\), bounded in the norm
\[
  \norm{u}_B=\sup_{t\in[0,1]}\left(\int_0^1 u(x,t)^2\dd x\right)^{1/2}\ ?
\]
\item[(b)] Under what conditions on \(f\) does the stability estimate
\[
  \norm{u}_B\le C\norm{f}_{L^2(0,1)}
\]
hold?
\item[(c)] Assuming \(u\in C^2(\overline{U_1})\), where \(U_1=(0,1)\times(0,1]\), is
uniqueness of solutions guaranteed?
\end{enumerate}

\begin{flushright}
(Error corrected 6-28-11.)
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