Abstract

We study a parabolic equation of the form ut−Δλu+a⋅∇λu=up inRN×R, and a parabolic system {ut−Δλu+a⋅∇λu=vput−Δλv+a⋅∇λv=uqinRN×R, where p is a real number, a(x) is a smooth vector field, Δλ is a strongly degenerate operator given by Δλ=∑i=1N∂xi(λ2i∂xi) and ∇λ is the gradient operator associated to Δλ. Under some general condition of λi introduced in Kogoj and Lanconelli [Nonlinear Anal 75: 4637–4649, 2012], we establish a Liouville type theorem for positive supersolutions to the problems above. In particular, we compute explicitly the critical exponent depending on both the homogeneous dimension of RN associated to Δλ and the behavior of the advection term a at infinity. This critical exponent is sharp in the case of Laplace operator without advection term according to [15].