Elliptic gradient estimates and Liouville theorems for a weighted nonlinear parabolic equation

Abstract

Let $(M^N,g,e^{-f}dv)$ be a complete smooth metric measure space with $\infty$-Bakry-Émery Ricci tensor bounded from below. We derive elliptic gradient estimates for positive solutions of a weighted nonlinear parabolic equation \begin{align*} \displaystyle \Big(Δ_f - \frac{\partial}{\partial t}\Big) u(x,t) +q(x,t)u^α(x,t) = 0, \end{align*} where $(x,t)\in M^N\times (-\infty ,\infty )$ and $\alpha$ is an arbitrary constant. As Applications we prove a Liouville-type theorem for positive ancient solutions and Harnack-type inequalities for positive bounded solutions.

18 pages

References in corpus (1)

• Gradient estimates for a nonlinear diffusion equation on complete manifolds