Gradient Estimates for a Nonlinear Parabolic Equation on Complete Smooth Metric Measure Spaces

Abstract

Let $$(M,g,e^{-\\phi } d\\nu )$$\n be a complete smooth metric measure space with the m-Bakry–Emery Ricci curvature bounded from below. In the first part of this article, we establish a local Hamilton-type gradient estimate for positive solutions to a nonlinear parabolic equation $$\\begin{aligned} (\\Delta _{\\phi }-q-\\partial _{t})u=au(\\ln u)^{\\alpha } \\end{aligned}$$\n on $$(M,g,e^{-\\phi } d\\nu )$$\n , where q(x,\xa0t) is a smooth space-time function and a, $$\\alpha $$\n are real constants. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. In the second part of this article, we obtain a refined global gradient estimates for the positive solutions to the above nonlinear parabolic equation with a smooth function q(x) and $$\\alpha =1$$\n . As its application, we derive a uniform bound for the solutions to the corresponding elliptic equation. Our results generalize some recent works on this direction.