Critical Exponents of Quasilinear Parabolic Equations

In this paper we study the critical exponents of the Cauchy problem in Rn of the quasilinear singular parabolic equations: ut = div ∇u m−1∇u + ts x σup, with non-negative initial data. Here s ≥ 0 n − 1 / n + 1 < m < 1 p > 1 and σ > n 1 − m − 1 + m + 2s . We prove that pc ≡ m + 1 + m + 2s + σ /n > 1 is the critical exponent. That is, if 1 < p ≤ pc then every non-trivial solution blows up in finite time, but for p > pc , a small positive global solution exists