  Constrained gradient flows are studied in fracture mechanics to describe
strongly irreversible (or unidirectional) evolution of cracks. The present
paper is devoted to a study on the long-time behavior of non-compact orbits of
such constrained gradient flows. More precisely, traveling wave dynamics for a
one-dimensional fully nonlinear Allen-Cahn type equation involving the
positive-part function is considered. Main results of the paper consist of a
construction of a one-parameter family of degenerate traveling wave solutions
(even identified when coinciding up to translation) and exponential stability
of such traveling wave solutions with some basin of attraction, although they
are unstable in a usual sense.
