Nonlinear Analysis-theory Methods & Applications | 2019
Gradient estimates for a nonlinear parabolic equation on smooth metric measure spaces
Abstract
Abstract Let ( M n , g , e − ϕ d v ) be a smooth metric measure space. In this paper, we derive a series of gradient estimates and a Harnack inequality for positive solutions of a nonlinear parabolic partial differential equation ( Δ ϕ − ∂ t ) u = q u + a u ( ln u ) α in M n × ( − ∞ , + ∞ ) , where q ∈ C 2 , 1 ( M n × ( − ∞ , + ∞ ) ) and a , α ∈ R .