Acta mathematica scientia, Series B >
GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS
Received date: 2014-10-13
Revised date: 2015-06-24
Online published: 2016-04-25
Supported by
This work is partially supported by the National Science Foundation of China (41275063 and 11401575).
In this article, we will derive local elliptic type gradient estimates for positive solutions of linear parabolic equations (Δ-∂/∂t)u(x, t)+q(x, t)u(x, t)=0 and nonlinear parabolic equations (Δ-∂/∂t)u(x, t)+h(x, t)up(x, t)=0(p>1) on Riemannian manifolds. As applications, we obtain some theorems of Liouville type for positive ancient solutions of such equations. Our results generalize that of Souplet-Zhang ([1], Bull. London Math. Soc. 38(2006), 1045-1053) and the author ([2], Nonlinear Anal. 74(2011), 5141-5146).
Xiaobao ZHU . GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS[J]. Acta mathematica scientia, Series B, 2016 , 36(2) : 514 -526 . DOI: 10.1016/S0252-9602(16)30017-0
[1] Souplet P, Zhang Q. Sharp gradient estimate and Yau's Liouville theorem for the heat equation on noncompact manifolds. Bull London Math Soc, 2006, 38:1045-1053
[2] Zhu X B. Gradient estimates and Liouville theorems for nonlinear parabolic equations on noncompact Riemannian manifolds. Nonlinear Anal, 2011, 74:5141-5146
[3] Li P, Yau S T. On the parabolic kernel of the Schrödinger operator. Acta Math, 1986, 156:153-201
[4] Li J Y. Gradient estimates and Harnack inequalities for nonlinear parabolic and nonlinear elliptic equations on Riemannian manifolds. J Funct Anal, 1991, 100:233-256
[5] Hamilton R S. A matrix Harnack estimate for the heat equation. Comm Anal Geom, 1993, 1:113-126
[6] Kotschwar B L. Hamilton's gradient estimate for the heat kernel on complete manifolds. Proc Amer Math Soc, 2007, 135(9):3013-3019
[7] Bailesteanu M, Cao X D, Pulemotov A. Gradient estimates for the heat equation under Ricci flow. J Funct Anal, 2010, 258:3517-3542
[8] Wu J. Gradient estimates for a nonlinear diffusion equation on complete manifolds. J Partial Differ Equ, 2010, 23(1):68-79
[9] Yang Y Y. Gradient estimates for a nonlinear parabolic equation on Riemannian manifolds. Proc Amer Math Soc, 2008, 136:4095-4102
[10] Chen L, Chen W Y. Gradient estimates for a nonlinear parabolic equation on complete non-compact Riemannian manifolds. Ann Global Anal Geom, 2009, 35(4):397-404
[11] Ma L, Zhao L, Song X. Gradient estimate for the degenerate parabolic equation ut=ΔF(u)+H(u) on manifolds. J Differential Equations, 2008, 244:1157-1177
[12] Ruan Q H. Elliptic-type gradient estimate for Schrödinger equations on noncompact manifolds. Bull Lond Math Soc, 2007, 39(6):982-988
[13] Wang L F. Elliptic type gradient estimates for the p-Laplace Schrödinger heat equation. Acta Math Sinica (Chin Ser), 2010, 53(4):643-654
[14] Wang M. Liouville theorems for the ancient solution of heat flows. Proc Amer Math Soc, 2010, 139(10):3491-3496
[15] Xu X. Gradient estimates for the degenerate parabolic equation ut=ΔF(u) on manifolds and some Liouville-type theorems. J Differential Equations, 2012, 252(2):1403-1420
[16] Yang Y Y. Gradient estimates for the equation Δu+cu-α=0 on Riemannian manifolds. Acta Math Sin (Engl Ser), 2010, 26(6):1177-1182
[17] Zhu X B. Hamilton's gradient estimates and Liouville theorems for fast diffusion equations on noncompact Riemannian manifolds. Proc Amer Math Soc, 2011, 139:1637-1644
[18] Zhu X B. Hamilton's gradient estimates and Liouville theorems for porous medium equations on noncompact Riemannian manifolds. J Math Anal Appl, 2013, 402(1):201-206
[19] Calabi E. An extension of E. Hopf's maximum principle with an application to Riemannian geometry. Duke Math J, 1958, 25:45-56
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