Math 6410 • ODE

Let \(D\subset\R^d\) be a bounded domain with smooth boundary \(\partial D\). Assume that
the complete, orthonormal set of eigenfunctions \(\{\phi_n\}_{n=1}^{\infty}\) for the
negative Dirichlet Laplacian on \(D\) is known, in other words,
\[
  -\Delta\phi_n=\lambda_n\phi_n,\quad \text{in }D,\ n=1,2,3,\ldots,
\]
with \(\phi_n(x)=0\) for \(x\in\partial D\), and \(\int_D\phi_n^2\dd x=1\).
\begin{enumerate}
\item[(a)] Given \(f\in L^2(D)\), find a closed-form series solution \(u(x,t)\) for the
initial boundary value problem for the heat equation
\[
\begin{aligned}
  u_t-\Delta u&=0,\quad t>0,\ x\in D\\
  u(x,t)&=0,\quad x\in\partial D,\\
  u(x,0)&=f,\quad x\in D.
\end{aligned}
\]
\item[(b)] Show that there exists a constant \(C\) such that
\(\left\lvert\int_D u(x,t)\phi_n(x)\dd x\right\rvert\le Ce^{-\lambda_n t}\),
for all \(n\). Find \(\lim_{t\to0}u(x,t)\).
\item[(c)] Give an example (the case \(d=1\) will suffice) which shows that the heat equation
does \textit{not} exhibit finite propagation speed.
\end{enumerate}