The Critical Exponent of Doubly Singular Parabolic Equations

In this paper we study the Cauchy problem of doubly singular parabolic equations ut = div ∇u σ ∇um + t x u with non-negative initial data. Here −1 < σ ≤ 0, m > max 0 1 − σ − σ + 2 /N satisfying 0 < σ +m ≤ 1, p > 1, and s ≥ 0. We prove that if θ > max − σ + 2 , 1 + s N 1 − σ − m − σ + 2 , then pc = σ +m + σ +m− 1 s + σ + 2 1+ s + θ /N > 1 is the critical exponent; i.e, if 1 < p ≤ pc then every non-trivial solution blows up in finite time. But for p > pc a positive global solution exists. © 2001 Academic Press