  We investigate slowly converging solutions for non-linear evolution equations
of elliptic or parabolic type. These equations arise from the study of isolated
singularities in geometric variational problems. Slowly converging solutions
have previously been constructed assuming the Adams-Simon positivity condition.
In this study, we identify a necessary condition for slowly converging
solutions to exist, which we refer to as the Adams-Simon non-negativity
condition. Additionally, we characterize the rate and direction of convergence
for these solutions. Our result partially confirms Thom's gradient conjecture
in the context of infinite-dimensional problems.
