  We show that all kinds of biasing of cosmological phase transitions produce
qualitatively new type of domain wall networks. The biased networks consist of
compact, finite size, bag-like wall structures and exhibit a generic
instability. The surface of biased networks disappears exponentially fast after
a limited period of scaling. We argue that fluctuations of the background make
the network unstable even in the case of the ``symmetric on the average''
initial distribution. We observe that the variation in parameters of the
potential, like its hight, can influence the lifetime of the wall network,
contrary to the standard beliefs.
