  In this paper, we study a numerical method for the solution of partial
differential equations on evolving surfaces. The numerical method is built on
the stabilized trace finite element method (TraceFEM) for the spatial
discretization and finite differences for the time discretization. The TraceFEM
uses a stationary background mesh, which can be chosen independent of time and
the position of the surface. The stabilization ensures well-conditioning of the
algebraic systems and defines a regular extension of the solution from the
surface to its volumetric neighborhood. Having such an extension is essential
for the numerical method to be well-defined. The paper proves numerical
stability and optimal order error estimates for the case of simplicial
background meshes and finite element spaces of order $m\ge1$. For the algebraic
condition numbers of the resulting systems we prove estimates, which are
independent of the position of the interface. The method allows that the
surface and its evolution are given implicitly with the help of an indicator
function. Results of numerical experiments for a set of 2D evolving surfaces
are provided.
