Life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values

In this paper, we investigate the behavior of the positive solution of the following Cauchy problem u t − div ( | ∇ u m | p − 2 ∇ u m ) = u q with initial value decaying at infinity, and give a new secondary critical exponent for the existence of global and nonglobal solutions. Furthermore, the large time behavior and the life span of solutions are also studied.