Elliptic-type gradient estimates under integral Ricci curvature bounds

Abstract

Let ( M n , g ) (\mathbf {M}^n, g) be an n n -dimensional complete Riemannian manifold. We prove that for any p > n 2 p>\frac {n}{2} , when k ( p , 1 ) k(p,1) is small enough, two certain elliptic-type gradient estimates hold for any positive solutions of the equation u t = Δ u + a u log ⁡ u \begin{equation*} u_{t}=\Delta u+au\log u \end{equation*} on geodesic balls B ( O , r ) B(O,r) in M n \mathbf {M}^n with 0 > r ≤ 1 0>r\leq 1 . Here the assumption that k ( p , 1 ) k(p,1) is small allows the situation where the manifold is collapsing. As applying, two forms of elliptic type gradient estimates to the heat equation on M n M^n under integral curvature conditions are obtained. Besides, each of the two forms of gradient estimate has its own advantages and differences (see Remark 4).