Math 6410 • ODE

For each \(n\in\N\), consider the Cauchy problem
\[
\begin{aligned}
  -\Delta u_n &= 0, && \text{in } U,\\
  u_n &= \frac{1}{n^2}\sin nx, && \text{on } \{(x,y):y=0\},\\
  \frac{\partial u_n}{\partial y} &= \frac{1}{n}\sin nx,
    && \text{on } \{(x,y):y=0\},
\end{aligned}
\]
where \(U=\{(x,y):0<y<1\}\). Find a sequence \(\{u_n\}\) of solutions to these problems,
prove that \(\{u_n\}\) does not converge to zero, and explain why this implies that the
Cauchy problem above is not ``well posed''.