The Critical Exponent of Doubly Singular Parabolic Equations1

In this paper we study the Cauchy problem of doubly singular parabolic equations ut = div ∇u σ ∇um + ts x θup 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.