本文選題：Ricci流 + 熱方程　； 參考：《中國(guó)科學(xué)技術(shù)大學(xué)》2017年博士論文

【摘要】：本文主要研究Ricci流下幾類(lèi)非線(xiàn)性?huà)佄锓匠陶獾奶荻裙烙?jì)以及Harnack不等式。主要研究?jī)?nèi)容包括:(1)將Li-Yau對(duì)流形上熱方程的梯度估計(jì)推廣到Ricci流下非線(xiàn)性?huà)佄锓匠?并建立相關(guān)的Harnack不等式;(2)沿Ricci流下,將Li-Yau估計(jì)中的常數(shù)α推廣到滿(mǎn)足一定系統(tǒng)的函數(shù)α(t);(3)作為滿(mǎn)足一定系統(tǒng)的特殊情形,我們給出Li-Yau型,Hamilton型,等梯度估計(jì)。(4)研究了黎曼流形上非線(xiàn)性?huà)佄锓匠?當(dāng)α ≤ 0和α ≥ 1時(shí),證明了 Hamilton橢圓型梯度估計(jì)和Liouville型定理,補(bǔ)充了 Zhu對(duì)0α1的結(jié)果。另外,導(dǎo)出了 Li-Yau型梯度估計(jì)和Harnack不等式,進(jìn)而將Li和Xu的結(jié)果推廣到非線(xiàn)性?huà)佄锓匠獭?5)研究了黎曼流形上介質(zhì)穿透型方程的Hamilton型梯度估計(jì)和Liouville型定理,從而進(jìn)一步簡(jiǎn)化了 Souplet和Zhang關(guān)于熱方程的結(jié)果。
[Abstract]:In this paper, the gradient estimation and Harnack inequality of positive solutions for some nonlinear parabolic equations under Ricci flow are studied. The main research contents include: (1) the gradient estimation of the thermal equation on the Li-Yau convection is extended to the nonlinear parabolic equation under the Ricci flow, and the related Harnack inequality is established. In this paper, the constant 偽 in Li-Yau estimator is extended to the function 偽 t 3 of a certain system. As a special case of satisfying a certain system, we give the Li-Yau type Hamiltonian type and equal gradient estimator .4) We study the nonlinear parabolic equations on Riemannian manifolds. When 偽 鈮，

本文編號(hào)：1969800