Gradient estimates for a nonlinear parabolic equation with diffusion on complete noncompact manifolds

Abstract

The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation: \frac{{\partial u}}{{\partial t}} = \Delta u - b(x,t)u^\sigma on complete noncompact manifolds with Ricci curvature bounded from below, where 0 < σ < 1 is a real constant, and b(x, t) is a function which is C ² in the x-variable and C ¹ in the t-variable.

Keywords

Gradient estimates / Positive solutions / Harnack inequality

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