Gradient estimates and Harnack inequalities for a Yamabe-type parabolic equation under the Yamabe flow (Q2037539)

The gradient estimate is an essential tool in understanding solutions of nonlinear partial differential equations from geometry. In 1980s, Li and Yau proved the well-known Li-Yau type estimate to the heat equation \((\Delta-\partial_t)u = 0\). Based on this, they deduced Harnack inequalities. The Harnack inequality was also applied to the Ricci flow by Hamilton and played an important role in solving the Poincaré conjecture. Hamilton proved an elliptic type gradient estimate, which is known as the Hamilton type gradient estimate, and showed that the Harnack estimate of Li and Yau is the trace of a full matrix inequality. Kotschwar later generalized the Hamilton type gradient estimate to the noncompact case. In 2018, Dung gave a global Hamilton type gradient estimate for a Yamabe type parabolic equation of \((\Delta-\partial_t)u = + au + bu^\alpha\) on Riemannian manifolds with the Ricci curvature bounded from below. In this paper, the author extends the previous results deriving a series of gradient estimates and Harnack inequalities for positive solutions of a Yamabe-type parabolic partial differential equation \[(\Delta- \partial_t)u = pu + qu^{a+1}\] under the Yamabe flow. Here \(p, q \in C^{2,1} (M^n \times [0, T])\) and \(a\) is a positive constant.