  Cosmic domain walls are harmless, provided that their tension decreases with
expansion of the Universe. This setup can be realized, if the scale of
spontaneous symmetry breaking is induced dynamically through the interaction
with hot primordial plasma. In that case, the domain wall tension can attain
large values in the early Universe without any conflict with observations.
Owing to the large initial tension, these topological defects may serve as a
powerful source of gravitational waves. We make a preliminary estimate of the
gravitational wave spectrum and argue that it is distinct from the spectrum
produced by other sources, in particular by domain walls of a constant tension.
The resulting gravitational wave signal is in the range accessible by Einstein
Telescope, DECIGO, TianQin, LISA, IPTA, or SKA, if the field constituting the
domain walls is very feebly coupled with hot primordial plasma and has tiny
self-interactions. In particular, one can consider this field for the role of
Dark Matter. We discuss various Dark Matter production mechanisms and
properties of the emitted gravitational waves associated with them. We find
that the conventional freeze-out and freeze-in mechanisms lead to large and
perhaps unobservable frequency of gravitational waves. However, the Dark Matter
production is also possible at the second order phase transition leading to the
domain wall formation or at the inverse phase transition, when the domain walls
get dissolved eventually. In both cases, there is essentially no lower bound on
the frequency of emitted gravitational waves.
