How to tell if you have specified sufficient initial data for a differential equation? I recently learnt that the following 'wave equation' is not well-posed $$
\begin{cases}
\partial_{tt}u=\partial_{xx} u, & (0,1)\times\mathbb R\\
u(t,0)=u(t,1)=0,&t\in [0,1]
\end{cases}
$$
since the solution will not be unique. I was told that in this case it is sufficient that one specifies $\partial_t u(0,0)$ and $\partial_t u(0,1)$ in order to have a unique solution and I can understand the proof. But, could someone please explain to me what's going on here morally? Why is it that specifying only $u(\cdot,0)$ and $u(\cdot,1)$ is not enough, but further specifying the derivatives then is enough? I'd love an explanation that allows me to have a feeling for more general equations as to what would constitute sufficient initial data.

If that's too vague, consider the concrete example: is the following heat-type equation well-posed 
$$
\begin{cases}
\partial_{t}u+\triangle^2u=0, & (0,\infty)\times\mathbb R^d\\
u(0,x)=f(x)\in C^\infty_c(\mathbb R^d\to\mathbb R)
\end{cases}
$$
provided we seek only solutions with sub-exponential growth (as with the usual heat equation) or do I need to specify more information about the derivatives of the solution at $t=0$ or something else entirely? And, how could you tell either way?
 A: Let's think of the function spaces as vector spaces, and of the linear equation (ODE or PDE) as being of this form: find $u$ that satisfies
$$
Au = f,
$$
where $f$ is a known function, and $A$ a known linear operator. 
The solution $u$ will fail to be unique if $A$ has a nontrivial kernel.  I.e., if $w$ lies in that kernel (i.e., if $Aw = 0$), and if $u$ is a solution, then so is $(u+w)$.
The boundary conditions are there to restrict the subspace (of the domain of $A$) from which we are allowed to take $u$.  If the intersection of this subspace with the kernel of $A$ is the zero vector space, we will have uniqueness.
A: When you can using separation of variables to find solutions, it often helps in sorting out the conditions needed for solutions. For example, in your first equation, $u(t,x)=T(t)X(x)$ is a solution of the equation, where $x\in\mathbb{R}^{d}$ if there is a parameter $\lambda$ such that
$$
                     \frac{T''(t)}{T(t)} = \lambda = \frac{\Delta X(x)}{X(x)}
$$
Because of the conditions $T(0)=0=T(1)$, the parameter $\lambda$ is determined by to $\lambda=-n^2\pi^2$ where $n=1,2,3\cdots$; the solutions $T$ are
$$
           T_n(t) = \sin(n\pi t).
$$
There are various ways of constructing solutions of
$$
             \Delta X = -n^2\pi^2 X.
$$
For example, if $\mu_n$ is any finite complex measure on $|\xi|=n\pi$ (which is a sphere in $\mathbb{R}^{d}$) then the following is such a solution:
$$
       X_{n,\mu}(x) = \int_{|\xi|=n\pi}e^{i\xi\cdot x}d\mu_n(\xi).
$$
So there are infinitely many solutions dictated by choices of constants $A_n$ and measures $\mu_n$:
$$
                u(x,t) = \sum_{n=1}^{\infty} A_n \sin(n\pi t)\int_{|\xi|=n\pi}e^{i\xi\cdot x}d\mu_n(\xi)
$$
