Reference : Harnack estimates for nonlinear parabolic equations under the Ricci flow
 Document type : E-prints/Working papers : First made available on ORBilu Discipline(s) : Physical, chemical, mathematical & earth Sciences : Mathematics To cite this reference: http://hdl.handle.net/10993/33520
 Title : Harnack estimates for nonlinear parabolic equations under the Ricci flow Language : English Author, co-author : Li, Yi [University of Luxembourg > Faculty of Science, Technology and Communication (FSTC) > Mathematics Research Unit >] Zhu, Xiaorui [China Maritime Police Academy] Publication date : 2017 Number of pages : 40 Peer reviewed : No Keywords : [en] Harnack estimate ; Ricci flow ; Geometric flow Abstract : [en] In this paper, we consider first the Li-Yau Harnack estimates for a nonlinear parabolic equation $\partial_{t}u=\Delta_{t}u-qu -au(\ln u)^{\alpha}$ under the Ricci flow, where $\alpha>0$ is a constant. To extend these estimates to a more general situation, in the second part, we consider the gradient estimates for a positive solution of the nonlinear parabolic equation $\partial _{t}u=\Delta _{t}u+hu^{p}$ on a Riemannian manifold whose metrics evolve under the geometric flow $\partial _{t}g(t)=-2S_{g(t)}$. To obtain these estimates, we introduce a quantity $\underline{\boldsymbol{S}}$ along the flow which measures whether the tensor $S_{ij}$ satisfies the second contracted Bianchi identity. Under conditions on ${\rm Ric}_{g(t)}, S_{g(t)}$, and $\underline{\boldsymbol{S}}$, we obtain the gradient estimates. Target : Researchers ; Professionals ; Students Permalink : http://hdl.handle.net/10993/33520 FnR project : FnR ; FNR7628746 > Anton Thalmaier > GEOMREV > Geometry Of Random Evolutions > 01/03/2015 > 28/02/2018 > 2014

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