I recently read this on Wikipedia, from here.
They belong to the class of systems with the functional state, i.e. partial differential equations (PDEs) which are infinite dimensional, as opposed to ordinary differential equations (ODEs) having a finite dimensional state vector.
What does this mean? Why are PDEs infinite and ODEs finite?
Let's take the Cauchy problem $y' = Ay$ with $y(0) = {y_0}$, with $A$ is an operator. When we talk about ODE, $A$ is aways a matrix which acts from $R^n$ to $R^m$ (finite dimension operator), for example dynamical system, but in pde theory the operator $A$ is an operator who defined on infinite dimensional space for example Laplacian, derevative operator, elliptique operator are always defined on some functional spaces ( Soboev spaces, $C^m$ ....etc).