  In this paper, we study a doubly nonlinear parabolic equation arising from
the gradient flow for p-Sobolev type inequality, referred as p-Sobolev flow. In
the special case p=2 our theory includes the classical Yamabe flow on a bounded
domain in Euclidean space. Our main aim is to prove the global existence of the
p-Sobolev flow together with its qualitative properties.
