Backwards heat equation (stability analysis) Problem

Consider the backwards heat equation of the form 
  $$ \left\{ \begin{aligned} u_{t} &= \lambda^2 u_{xx}, & x\in[0,L], \quad t\in[0,T]\\ u(0,t) &= u(L,t) = 0 \\ u(x,T) &= f(x), \end{aligned} \right.\tag{*}\label{*}$$
  Establish whether solution is unique and analyze its stability.

Attempt of proving uniqueness
My attempt to prove uniqueness is provided in this post.
Attempt of (dis)proving stability
The general solution of $\eqref{*}$ is of the form 
$$
u(x,t) = \sum_{m=1}^{\infty} A_m \sin\bigg( \frac{\pi m }{L}\,x\bigg) \exp \Bigg(\!\!-\!\bigg(\frac{\pi m }{L}\bigg)^2  \lambda^2 \big(T -t \big) \Bigg)\\
A_m =  \frac{2}{L} \int_0^L \sin \!\bigg( \frac{\pi m }{L}\,x\bigg)\, f(x)\,dx
$$
I think the solution is not stable in $L^p$ sense, so I need to come up with a good counterexample of the sequence of initial data  $f_n(x)\to f(x)$ converging in $L_p$, so that it would not be difficult to show that corresponding solutions do not converge in $L_p$. 
Could anyone propose such an example?

Use a sequence of harmonics with increasing frequency and decreasing amplitude: 
$$f_n(x)= A_n \sin \left(\frac{\pi n}{L} x\right)$$
As long as $A_n\to 0$, the   values at time $T$ tend to zero in the $L^p$ sense. On the other hand, the solution at time $t<T$ is 
$$f_n(x)= A_n e^{\lambda (\pi n/L)^2 (T-t)}\sin \left(\frac{\pi n}{L} x\right)$$
which, for most natural choices of $A_n$, blows up in $L^p$ norm.
