New critical exponents for a doubly singular parabolic equation. Issue 11 (18th August 2021) - Zheng, Yadong; Fang, Zhong Bo
ABSTRACT: This paper deals with the Cauchy problem for a doubly singular parabolic equation with nonlocal inner source 1 u t = d i v ( | ∇ u m | p − 2 ∇ u m ) + ∥ u ∥ L q ( R N ) r − 1 u s + 1, ( x, t ) ∈ R N × ( 0, T ), where N ≥ 1, 1 − ( ( 1 + m ) / ( 1 + m N ) ) < m ( p − 1 ) < 1, 0 < m ≤ 1, q >1, r ≥ 1, 0 ≤ s < p / N + m ( p − 1 ) − 1 and r + s >1. We first obtain a new critical Fujita exponent by virtue of the auxiliary function method and the forward self-similar solution, and then determine the second critical exponent to classify global and non-global solutions of the problem in the coexistence region via the decay rates of an initial data at spatial infinity. Moreover, the large time behavior of global solution and the life span of non-global solution are derived.