Triviality of bounded solutions and gradient estimates for nonlinear f-heat equations on complete smooth metric measure spaces

Abstract

Some qualitative properties of positive smooth solutions to a generalized nonlinear parabolic equation involving f-Laplacian (Lf)(Lf−q(x,t)−∂∂t)w(x,t)=G(w(x,t)), are discussed on Mf×(−∞,+∞), where Mf is a complete smooth metric measure space (with or without boundary), the potential function q(x,t) is smooth at least C1 in x and C0 in t, and G(w(x,t)) is a nonlinear smooth sourcing term. Local and global type space only (elliptic type) gradient estimates are established for this equation under the condition that Bakry-Émery Ricci curvature tensor is bounded from below. As an exploitation of the gradient estimates so derived we obtain a parabolic Harnack inequality and some Liouville type theorems for bounded ancient and eternal solutions. The approach adopted in this paper provides a unified treatment of a large class of nonlinear source terms. To demonstrate this further, cases where G(w)=awρ, a∈R, ρ∈(−∞,0]∪[1,+∞), G(w)=aw|log⁡w|γ,a∈R, a≠0, γ>1 and G(w)=aw(log⁡w)γa≠0, γ≥1 are considered as specific examples. We further show that all the obtained results also hold on weighted manifolds with compact boundary under some lower boundedness assumptions on mean curvature of the boundary and Bakry-Émery Ricci curvature tensor.