Gradient estimates for a nonlinear parabolic equation and Liouville theorems

Abstract

We establish local elliptic and parabolic gradient estimates for positive smooth solutions to a nonlinear parabolic equation on a smooth metric measure space. As applications, we determine various conditions on the equation's coefficients and the growth of solutions that guarantee the nonexistence of nontrivial positive smooth solutions to many special cases of the nonlinear equation. In particular, we apply gradient estimates to discuss some Yamabe-type problems of complete Riemannian manifolds and smooth metric measure spaces.

final version, accepted by Manuscripta Math

References in corpus (3)

• Gradient estimates for some $f$-heat equations driven by Lichnerowicz's equation on complete smooth metric measure spaces
• Differential Harnack Estimates for Fisher's Equation
• Harnack Estimate for the Endangered Species Equation

Cited by in corpus (4)

• Gradient estimates for a nonlinear parabolic equation with Dirichlet boundary condition
• Gradient Einstein-type warped products: rigidity, existence and nonexistence results via a nonlinear PDE
• Harnack Estimate For Positive Solutions to a Nonlinear Equation Under Geometric Flow
• Gradient Estimates And Liouville Theorems For A Class of Nonlinear Elliptic Equations