  We study integrability of the derivative of solutions to a singular
one-dimensional parabolic equation with initial data in $W^{1,1}$. In order to
avoid additional difficulties we consider only the periodic boundary
conditions. The problem we study is a gradient flow of a convex, linear growth
variational functional. We also prove a similar result for the elliptic
companion problem, i.e. the time semidiscretization.
