Citation Context

...lds. In 2007, B. Kotschwar ([11]) used Hamilton’s estimate and obtained a global gradient estimate for heat kernels on complete noncompact manifolds. In 2010, M. Bailesteanu, X. Cao and A. Pulemotov (=-=[2]-=-) generalized Souplet and Zhang’s result to Ricci flow. Also in 2010, on complete noncompact Riemannian manifolds, J. Wu ([15]) obtained a localized Hamilton-type gradient estimate for the positive sm...

Citation Context

...ion under the Ricci flow. On the subject of differential Harnack inequalities for the linear heat equation coupled with the Ricci flow, there have been many important contributions; see, for example, =-=[3]-=-, [7], [10], [17], [22], [32], [40] and [44]. In recent years there has been increasing interest in the study of the nonlinear heat-type equations coupled with the Ricci flow. A nice example of a nonl...

Citation Context

...tive solutions to the conjugate heat equation under the Ricci flow on closed manifolds with nonnegative scalar curvature. Besides the above work, there were also many research papers (see for example =-=[1]-=-, [3], [5], [8], [11], [18], [28] and [30]). In order to make a clear statement of our Harnack inequalities, we need to recall some known results, which are more or less related to our results. In [9]...

Citation Context

...owledge about the solvability of boundary-value problems for (1.1) may help answer questions about boundary-value problems for the Ricci flow and the Einstein equation. Such questions were studied in =-=[27, 1, 15, 5, 2, 16, 25]-=- and other works. A large number still remain open. The present paper investigates a Dirichlet-type problem for equation (1.1) on M . Suppose M1 is a connected manifold with ∂M1 = ∅. It will be conven...

Citation Context

...ith respect to g. Our proposition touches upon the question of the behavior of the Ricci flow on manifolds with convex boundary. This question is natural, and it is related to some of the material in =-=[22, 23, 11, 4]-=-. Among other things, our proposition implies that, if ∂M is convex with respect to ĝ, we can find T > 0 and a solution g of (1.4)–(1.5) on M × [0, T ) such that ∂M remains convex with respect to g. ...

Citation Context

...ods applied to parabolic equations were first studied by Li and Yau in their celebrated paper [12]. They have been applied successfully to the setting of various geometric flows, for more details see =-=[1, 2, 3, 4, 5, 6, 9, 11, 13]-=- and the references therein. See the paper [10] for similar applications. In this paper we will drive various gradient estimates for the following nonlinear parabolic equation with potential( ∂ ∂t −∆+...

Citation Context

..., 26, 5, 8], but it is not yet deeply understood. In fact, the majority of available results concern the special case where λ = 0 (i.e., the boundary is totally geodesic). We invite the reader to see =-=[3]-=- for a discussion about letting the parameter λ depend on t. Aside from the discussion, that paper contains two gradient estimates for the heat equation under (1.1)–(1.3). Further progress in the stud...