We consider the Cauchy problem to a class of fast-diffusion non-Newtonian filtration equations. Besides the usual degeneracy in the fast-diffusion non-Newtonian filtration, the equation is degenerate or singular at infinity, depending on the sign of the parameter related to the coefficient of diffusion. Fujita type theorems are established and the critical Fujita exponent is determined. Specially, we also prove that the nontrivial solution blows up in a finite time on the critical situation.
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