I have been given the following two definitions:
1) $D^ku$ is the set of all derivatives of order k of u
2) Let $\Omega$ be a non-empty subset of Euclidean space $\mathbb{R}^N.$ An expression of the form $$F(D^ku(x), D^{k-1}u(x),\dots, Du(x), u(x), x) = 0$$ for $x \in \Omega$ is called a kth order PDE where $F:\mathbb{R}^{N^k} \times \mathbb{R}^{N^{k-1}} \times \dots \times \mathbb{R}^N \times \mathbb{R} \times \Omega \to \mathbb{R}$ is given and $u: \Omega \to \mathbb{R}$ is unknown.
I am struggling to understand why an element of $D^ku$ is in $\mathbb{R}^{N^k}$. Also, what convention is used in writing $D^ku$?
It seems you might be working with Evans' book on PDE, and if you've never encountered multivariable calculus before, it might be a good idea to start with that first.
Take a simple example: $f(x,y) : \mathbb{R}^2 \rightarrow \mathbb{R}$. Here, $N = 2$. Then $Df = (f_x(x,y), f_y(x,y))$, where $f_x := \frac{\partial f}{\partial x}$, and ditto for $f_y$. Thus, $Df$, also sometimes denoted $\nabla f$, is an element of $\mathbb{R}^2$.
Next, $$ D^2 f = \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \end{pmatrix}.$$ This matrix is sometimes called the Hessian, and $D^2 f$ is in $R^{2 \times 2} = R^{2^2}$.
The pattern then continues, as we can take two different derivatives w.r.t. each of $f_{xx}, f_{yx}, f_{xy}, f_{yy}$. This yields eight numbers which we can put into a $2 \times 2 \times 2$ matrix "cube", which lives in $R^{2^3}$.
If you are indeed using Evans (your definition is almost verbatim from his first chapter), there is a nice appendix at the back of the book which explains all conventions and gives a calculus refresher.