  We study the evolution of cosmological domain walls in models with asymmetric
potentials. Our research goes beyond the standard case of spontaneous breaking
of an approximate symmetry. When the symmetry is explicitly broken the
potential exhibits nearly degenerate minima which can lead to creation of a
metastable network of domain walls. The time after which the network will decay
depends on the difference of values of the potential in minima, its asymmetry
around the maximum separating minima and the bias of initial distribution.
Using numerical lattice simulations we determine relative importance of these
factors on decay time of networks for generic potentials. We find that even
very small departures from the symmetric case lead to rapid decay of the domain
wall network. As a result creation of a long lasting network capable of
producing observable gravitational wave signals is much more difficult than
previously thought.
