I'm a bit stuck on how to approach this question.
Suppose someone treats one of the variables in a Laplace equation as time and tries to solve the evolution problem,
$$u_{tt}+u_{xx}=0\ (0<x<l\,,\ t>0)\,,\quad u(0,t)=u(l,t)=0\,,\quad u(x,0)=\phi(x)\,,\quad u_t(x,0)=\psi(x) $$
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).
I initially thought trying to solve with separation of variables so you have,
$$\frac{\ddot{T}(t)}{T(t)} = \frac{X''(x)}{X(x)} = -\lambda$$
So you end up with ,
$$T(t) = \sin(\sqrt{\lambda}t) + \cos(\sqrt{\lambda} t)$$ and $$X(x) = A e^{\sqrt{\lambda x}} + B e^{-\sqrt{\lambda x}}$$
and using the initial conditions on $X$ you end up with,
$$X(0) = A+B=0\implies B=-A$$
$$X(l) = A(e^{\sqrt{\lambda l}} - e^{-\sqrt{\lambda l}}) = 0$$
and since we want a non-trivial solution we solve for $\lambda$ and get that $\lambda = 0$ or imaginary, but I'm not sure if imaginary eigenvalues are allowed for this type of question.
Is this the correct way to approach it or should I try to use d'Alembert's formula where $c=i$?
I was going about it in the wrong way. Because the goal is just to show the instability it's enough to say that there is a solution
$$u(x,t)=A\sin(n x)e^{n t}$$
since
$$u_{tt} = -A n^2\sin(n x)e^{n t}\,,\quad u_{xx} = A n^2 \sin(n x)e^{n t}$$
and we can set $l=\pi$ for simplicity
Then if we let $A=\frac{1}{n}$
$$u(x,0)\rightarrow_{n\rightarrow \infty} 0$$ $$u(x,1)\rightarrow_{n\rightarrow\infty} \infty$$
which shows that since it is ill-posed with respect to the parameter $n$ so it has no continuous dependence on the data.