  We study doubly nonlinear parabolic equation arising from the gradient flow
for p-Sobolev type inequality, referred as p-Sobolev flow from now on, which
includes the classical Yamabe flow on a bounded domain in Euclidean space in
the special case p=2. In this article we establish a priori estimates and
regularity results for the $p$-Sobolev type flow, which are necessary for
further analysis and classification of limits as time tends to infinity.
