GRADIENT ESTIMATES AND LIOUVILLE THEOREMS FOR LINEAR AND NONLINEAR PARABOLIC EQUATIONS ON RIEMANNIAN MANIFOLDS

doi:10.1016/S0252-9602(16)30017-0

Abstract

Abstract:

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)u^p(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).

Key words: Gradient estimate, linear parabolic equation, nonlinear parabolic equation, Liouville type theorem

CLC Number:

35K55

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.

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