Math 6410 • ODE

\begin{enumerate}
\item Let \(H\) be a Hilbert space and let
\[
  A:D(A)\to H
\]
be a linear operator with domain \(D(A)\). State the Hille-Yosida-Phillips theorem giving
necessary and sufficient conditions for the unique solvability of the problem
\[
\begin{aligned}
  u'+Au&=0,\\
  u(0)&=u_0\in D(A).
\end{aligned}
\]
Be sure to define pertinent terms.

\item Apply the above result to the initial boundary value problem
\[
\begin{aligned}
  u_t-u_x&=0,\qquad t>0,\ x\in\R,\\
  u(0,x)&=g(x),\qquad x\in\R.
\end{aligned}
\]
Solve this problem explicitly.

Also state a result about the initial value problem
\[
\begin{aligned}
  u_t-u_x&=f(t,x),\qquad t>0,\ x\in\R,\\
  u(0,x)&=g(x),\qquad x\in\R.
\end{aligned}
\]
\end{enumerate}