Math 6410 • ODE

Let \(D\in\R^d\) be a bounded domain with smooth boundary \(\partial D\). Assume that that
a complete orthonormnal set of eigenfunctions \(\{\phi_n\}_{n=1}^\infty\) for the
Laplacian on \(D\) with Dirichlet boundary conditions are known, in other words
\begin{equation}
  -\Delta\phi_n=\lambda_n\phi_n,\qquad \text{in }D,\ n=1,2,3\ldots,
  \tag{6}
\end{equation}
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
\[
\begin{aligned}
  -u_t+\Delta u &= 0, && t>0,\ x\in D,\\
  u(x,t) &= 0, && x\in\partial D,\\
  u(x,0) &= 0, && x\in D.
\end{aligned}
\]
\item[(b)] Show that there is a constant \(c\) such that
\[
  \left\lvert\int_D u(x,t)\phi_n(x)\dd x\right\rvert \le c e^{-\lambda_n t}
\]
for all \(n\). Find \(\lim_{t\to\infty}u(x,t)\).
\item[(c)] Give an example (the case \(d=1\) will suffice) which shows that solutions of
the heat equation do not exhibit finite propagation speed.
\end{enumerate}