  This work is concerned with the gradient flow of absolutely $p$-homogeneous
convex functionals on a Hilbert space, which we show to exhibit finite ($p<2$)
or infinite extinction time ($p \geq 2$). We give upper bounds for the finite
extinction time and establish convergence rates of the flow. Moreover, we study
next order asymptotics and prove that asymptotic profiles of the solution are
eigenfunctions of the subdifferential operator of the functional. To this end,
we compare with solutions of an ordinary differential equation which describes
the evolution of eigenfunction under the flow. Our work applies, for instance,
to local and nonlocal versions of PDEs like $p$-Laplacian evolution equations,
the porous medium equation, and fast diffusion equations, herewith generalizing
many results from the literature to an abstract setting.
  We also demonstrate how our theory extends to general homogeneous evolution
equations which are not necessarily a gradient flow. Here we discover an
interesting integrability condition which characterizes whether or not
asymptotic profiles are eigenfunctions.
