Gradient estimate for a nonlinear heat equation on Riemannian manifolds

Abstract

In this paper, we derive a local Hamilton type gradient estimate for a nonlinear heat equation on Riemannian manifolds. As its application, we obtain a Liouville type theorem.

Global gradient estimates for a general type of nonlinear parabolic equations

“…Though several gradient estimate results have been obtained in different cases (see, e.g., [2,5,8,9,13,14,16,17,[23][24][25][26][27][28]), we provide here a general framework dealing, at once, with various nonlinearities of interest (as a matter of fact, a number of classical and recent results can be re-obtained as special cases of our general approach). Also, we will provide "global" (rather than "local") estimates that take into account the parabolic boundary behavior, thus improving the estimates when the data of the equation are particularly favorable.…”

Sharp gradient estimates for a heat equation in Riemannian manifolds

“…Therefore, our estimate is better than those in [Jia16,Wu17] for the case a ≤ 0. Moreover, the Liouville type results we obainned in the case a ≤ 0 for the equation (1.2) are also better than those in [DKN18, HM15, Li15, Jia16, Wu17].…”

Gradient estimates for some evolution equations on complete smooth metric measure spaces

“…where a, b ∈ R are fixed constants. To prove Theorem 1.3, let, as in [12] and [19], h(x, t) := u 1/3 (x, t).…”

“…They proved several differential Harnack inequalities and used them to derive bounds on gradient Ricci solitons. Recently, in [12], Jiang introduced a local Hamilton-type gradient estimate for (1.2) and obtained a Liouville-type theorem for bounded smooth solutions of (1.2). We refer the reader to [2], [4], [9], [12], and [22] for further references.…”