Gradient estimates and Harnack inequalities of a parabolic equation under geometric flow

Abstract

In this paper, we consider a manifold evolving by a general geometric flow and study parabolic equationWe establish space-time gradient estimates for positive solutions and elliptic type gradient estimates for bounded positive solutions of this equation. By integrating the gradient estimates, we derive the corresponding Harnack inequalities. Finally, as applications, we give gradient estimates of some specific parabolic equations.

Ultimate Bearing Capacity Analysis of CFRP-Strengthened Shield Segments Using Bonding Slip Behavior Experiments

“…To estimate the ultimate bearing capacity [34][35][36][37], maximum deflection, and flexural strength of damaged shield tunnel segment strengthening with CFRP, the tangential stress of the strengthening shield segments was analyzed as shown in Figure 16 [38][39][40]. The maximum and minimum values of the tangential stress can be calculated using the equations in Appendix B [41,42].…”

Aronson–Bénilan estimates for weighted porous medium equations under the geometric flow

“…Recently, many authors used similar techniques to prove gradient estimates and Harnack inequalities for positive solutions of parabolic equations under the geometric flows; see, for example, previous studies 20–26 . In 2014, B. Ma and J. Li 27 obtained Li–Yau type estimates for porous medium equation $$$ {u}_t=\Delta {u}^p $$$ in $$$ M\times \left(0,T\right] $$$ under the Ricci flow for $$$ p>1 $$$.…”

Gradient estimates for a weighted parabolic equation under geometric flow

“…Li-Yau type, Hamilton type, and Souplet-Zhang type gradient estimates have been obtained for other nonlinear parabolic equations on manifolds, for instance see [8,10,13,20,25,26,29,35] and the references therein. On the other hand, many authors used similar techniques to prove gradient estimates and Harnack inequalities for positive solutions of parabolic equations under the geometric flow, see for example [1,5,14,18,23,33,38].…”

“…In [10], Q. Chen and G. Zhao studied the equation (∆ − q − ∂ t )u = A(u) with a convection terms on a complete manifold with a fixed metric where A(u) is a function of C 2 in u. Then, in [38], G. Zhao obtained Li-Yau type and Hamilton type gradient estimates of equation (∆ − q − ∂ t )u = A(u) on Riemannian manifold evolving by the geometric flow. In [34], J. Y. Wu gave a local Li-Yau Type gradient estimate for the positive solutions to a nonlinear parabolic equation…”

“…In this paper, we establish some gradient estimates for bounded positive solution of (1.1) under the geometric flow (1.2), which are richer than [37,38]. In following we recall some basic definitions on an n-dimensional weighted Riemannian manifold (M, g, e −φ dv).…”