Math 6410 • ODE

Suppose someone treats one of the variables in the Laplace equation as time and
tries to solve the evolution problem
\[
  \begin{aligned}
    u_{tt}+u_{xx}&=0\quad(0<x<l,\ t>0),\\
    u(0,t)&=u(l,t)=0,\qquad
    u(x,0)=\phi(x),\qquad u_t(x,0)=\psi(x),
  \end{aligned}
\]
similar to the wave equation. Show that this problem has no continuous dependence
on data, even if the time \(t\) belongs to a finite interval \(0<t<T\) (\(T\) is a
positive constant).