  Four-dimensional cylindrically symmetric spacetimes with homothetic
self-similarity are studied in the context of Einstein's Theory of Gravity, and
a class of exact solutions to the Einstein-massless scalar field equations is
found. Their local and global properties are investigated and found that they
represent gravitational collapse of a massless scalar field. In some cases the
collapse forms black holes with cylindrical symmetry, while in the other cases
it does not. The linear perturbations of these solutions are also studied and
given in closed form. From the spectra of the unstable eigen-modes, it is found
that there exists one solution that has precisely one unstable mode, which may
represent a critical solution, sitting on a boundary that separates two
different basins of attraction in the phase space.
