Math 6410 • ODE

Let \(U\subset\R^n\) be open and bounded, with smooth boundary. For \(T>0\),
denote \(U_T=U\times(0,T]\). Let \(\Gamma_T=\overline{U_T}-U_T\) (closure taken in
\(\R^n\times\R\)). Recall that for solutions \(u(x,t)\) of
\[
\begin{aligned}
  u_t-\Delta u &= f, && \text{in } U_T,\\
  u &= g, && \text{on } \Gamma_T,
\end{aligned}
\]
we define the energy
\[
  e(t)=\int_U u^2(x,t)\,dx.
\]
\begin{enumerate}
\item[(a)] Use energy methods to show that the problem above has at most one solution
\(u\in C^{2,1}(U_T)\) (\(C^2\) in the \(x\) variables, and \(C^1\) in \(t\)).

\item[(b)] Assuming \(g=0\), and \(u\in C^{2,1}(U_T)\), prove that for \(0<t<T\),
\[
  e(t)\le
  \left(\int_0^t\left(\int_U f^2(x,s)\,dx\right)^{1/2}\,ds\right)^2.
\]
\end{enumerate}